?Inverted Pendulum Since the general solution of the linearized pendulum equation

Solutions in General For Problems 1-10, detennine tile general .solution and give the basis B = {y1, y2} for the solution space. \(y^{\prime \prime}+9 y=0\) ________________ Equation Transcription: Text Transcription: y''+9y=0

Solutions in General For Problems 1-10, detennine tile general .solution and give the basis B = {y1, y2} for the solution space. \(y^{\prime \prime}+y^{\prime}+y=0\) ________________ Equation Transcription: Text Transcription: y''+y'+y=0

Solutions in General For Problems 1-10, detennine tile general .solution and give the basis B = {y1, y2} for the solution space. \(y^{\prime \prime}-4 y^{\prime}+5 y=0\) ________________ Equation Transcription: Text Transcription: y''-4y'+5y=0

Solutions in General For Problems 1-10, detennine tile general .solution and give the basis B = {y1, y2} for the solution space. \(y^{\prime \prime}+2 y^{\prime}+8 y=0\) ________________ Equation Transcription: Text Transcription: y''+2y'+8y=0

Solutions in General For Problems 1-10, detennine tile general .solution and give the basis B = {y1, y2} for the solution space. \(y^{\prime \prime}+2 y^{\prime}+4 y=0\) ________________ Equation Transcription: Text Transcription: y''+2y'+4y=0

Solutions in General For Problems 1-10, detennine tile general .solution and give the basis B = {y1, y2} for the solution space. \(y^{\prime \prime}-4 y^{\prime}+7 y=0\) ________________ Equation Transcription: Text Transcription: y''-4y'+7y=0

Solutions in General For Problems 1-10, detennine tile general .solution and give the basis B = {y1, y2} for the solution space. \(y^{\prime \prime}-10 y^{\prime}+26 y=0\) ________________ Equation Transcription: Text Transcription: y''-10y'+26y=0

Solutions in General For Problems 1-10, detennine tile general .solution and give the basis B = {y1, y2} for the solution space. \(3 y^{\prime \prime}+4 y^{\prime}+9 y=0\) ________________ Equation Transcription: Text Transcription: 3y''+4y'+9y=0

Solutions in General For Problems 1-10, detennine tile general .solution and give the basis B = {y1, y2} for the solution space. \(y^{\prime \prime}-y^{\prime}+y=0\) ________________ Equation Transcription: Text Transcription: y''-y'+y=0

Solutions in General For Problems 1-10, detennine tile general .solution and give the basis B = {y1, y2} for the solution space. \(y^{\prime \prime}+y^{\prime}+2 y=0\) ________________ Equation Transcription: Text Transcription: y''+y'+2y=0

Initial· Value Problems Solve the IVPs in Problems 11-16. \(y^{\prime \prime}+4 y=0, \quad y(0)=1, \quad y^{\prime}(0)=1\) ________________ Equation Transcription: Text Transcription: y''+4y=0, y(0)=1, y'(0)=1

Initial· Value Problems Solve the IVPs in Problems 11-16. \(y^{\prime \prime}-4 y^{\prime}+13 y=0, \quad y(0)=1, \quad y^{\prime}(0)=1\) ________________ Equation Transcription: Text Transcription: y''-4y'+13y=0, y(0)=1, y'(0)=1

Initial· Value Problems Solve the IVPs in Problems 11-16. \(y^{\prime \prime}+2 y^{\prime}+2 y=0, \quad y(0)=1, \quad y^{\prime}(0)=0\) ________________ Equation Transcription: Text Transcription: y''+2y'+2y=0, y(0)=1, y'(0)=0

Initial· Value Problems Solve the IVPs in Problems 11-16. \(y^{\prime \prime}-y^{\prime}+y=0, \quad y(0)=0, \quad y^{\prime}(0)=1\) ________________ Equation Transcription: Text Transcription: y''-y'+y=0, y(0)=0, y'(0)=1

Initial· Value Problems Solve the IVPs in Problems 11-16. \(y^{\prime \prime}-4 y^{\prime}+7 y=0, \quad y(0)=0, \quad y^{\prime}(0)=-1\) ________________ Equation Transcription: Text Transcription: y''-4y'+7y=0, y(0)=0, y'(0)=-1

Initial· Value Problems Solve the IVPs in Problems 11-16. \(y^{\prime \prime}+2 y^{\prime}+5 y=0, \quad y(0)=1, \quad y^{\prime}(0)=-1\) ________________ Equation Transcription: Text Transcription: y''+2y'+5y=0, y(0)=1, y'(0)=-1

Working Backwards Write the standard form (equation (16) with leading coefficient an = I) of the nth order linear homogeneous differential equation with real coefficients whose roots are given in Problems 17-20. 3rd order, r = 1, 1 , 1

Working Backwards Write the standard form (equation (16) with leading coefficient an = I) of the nth order linear homogeneous differential equation with real coefficients whose roots are given in Problems 17-20. 3rd order, two of the roots arc r = 4, I - i

Working Backwards Write the standard form (equation (16) with leading coefficient an = I) of the nth order linear homogeneous differential equation with real coefficients whose roots are given in Problems 17-20. 3rd order, two of the roots are r = -2 + i, 2 + i

Working Backwards Write the standard form (equation (16) with leading coefficient an = I) of the nth order linear homogeneous differential equation with real coefficients whose roots are given in Problems 17-20. 4th order, three of the roots are 2, -2, 4 + i

Matching Problem For Problems 21-28. determine which graph of the particular solution shown in Fig. 4.3.9 matches each differential equation. \(y^{\prime \prime}-y^{\prime}=0\) ________________ Equation Transcription: Text Transcription: y''-y'=0

Matching Problem For Problems 21-28. determine which graph of the particular solution shown in Fig. 4.3.9 matches each differential equation. \(y^{\prime \prime}+y^{\prime}=0\) ________________ Equation Transcription: Text Transcription: y''+y'=0

Matching Problem For Problems 21-28. determine which graph of the particular solution shown in Fig. 4.3.9 matches each differential equation. \(y^{\prime \prime}+3 y^{\prime}+2 y=0\) ________________ Equation Transcription: Text Transcription: y''+3y'+2y=0

Matching Problem For Problems 21-28. determine which graph of the particular solution shown in Fig. 4.3.9 matches each differential equation. \(y^{\prime \prime}-5 y^{\prime}+6 y=0\) ________________ Equation Transcription: Text Transcription: y''-5y'+6y=0

Matching Problem For Problems 21-28. determine which graph of the particular solution shown in Fig. 4.3.9 matches each differential equation. \(y^{\prime \prime}+y^{\prime}+y=0\) ________________ Equation Transcription: Text Transcription: y''+y'+y=0

Matching Problem For Problems 21-28. determine which graph of the particular solution shown in Fig. 4.3.9 matches each differential equation. \(y^{\prime \prime}+y=0\) ________________ Equation Transcription: Text Transcription: y''+y=0

Matching Problem For Problems 21-28. determine which graph of the particular solution shown in Fig. 4.3.9 matches each differential equation. \(y^{\prime \prime}+4 y^{\prime}+4 y=0\) ________________ Equation Transcription: Text Transcription: y''+4y'+4y=0

Matching Problem For Problems 21-28. determine which graph of the particular solution shown in Fig. 4.3.9 matches each differential equation. \(y^{\prime \prime}-y^{\prime}+y=0\) ________________ Equation Transcription: Text Transcription: y''-y'+y=0

Euler’s Formula You can use the following process to justify Euler’s formula \(e^{i \theta}=\cos \theta+i \sin \theta\). (a) Write out explicitly the first dozen or so terms of the Maclaurin series (the Taylor expansion about the origin) given by \(e^{x}=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}\) (b) The series is valid for both real and complex numbers. Replace x by \(i \theta\) and write the expression for \(e^{i \theta}\). (c) Simplify the results by using the periodicity of powers of i: \(i^{0}=i^{4}=i^{8}=\cdots=1\), \(i^{1}=i^{5}=i^{9}=\cdots=i\), \(i^{2}=i^{6}=i^{10}=\cdots=-1\), \(i^{3}=i^{7}=i^{11}=\cdots=-i\). (d) Collect the real and imaginary terms. (e) Obtain Euler’s formula by recognizing the two Maclaurin series that appear in pan (d). ________________ Equation Transcription: Text Transcription: i^0=i^4=i^8= ... =1 i^1=i^5=i^9= ... =i i^2=i^6=i^10= ... =-1 i^3=i^7=i^11= ... =-i

Long-Term Behavior of Solutions Suppose that r1 and r2 are the characteristic roots for \(y^{\prime \prime}+b y^{\prime}+c y=0\), so the solutions is \(y(t)=c_{1} e^{r_{1} t}+c_{2} e^{r_{2} t}\). For Problems 30-35, discuss the long-term solution behaviors for the given r1, r2 combinations. Assume \(\beta \neq 0\). \(r_{1}<0, r_{2}<0\) ________________ Equation Transcription: Text Transcription: y''+by'+cy=0 y(t)=c_1e^r_1t+c_2e^r_2t Beta neq 0 r_1<0, r_2<0

Long-Term Behavior of Solutions Suppose that r1 and r2 are the characteristic roots for \(y^{\prime \prime}+b y^{\prime}+c y=0\), so the solutions is \(y(t)=c_{1} e^{r_{1} t}+c_{2} e^{r_{2} t}\). For Problems 30-35, discuss the long-term solution behaviors for the given r1, r2 combinations. Assume \(\beta \neq 0\). \(r_{1}<0, r_{2}=0\) ________________ Equation Transcription: Text Transcription: y''+by'+cy=0 y(t)=c_1e^r_1t+c_2e^r_2t Beta neq 0 r_1<0, r_2=0

Long-Term Behavior of Solutions Suppose that r1 and r2 are the characteristic roots for \(y^{\prime \prime}+b y^{\prime}+c y=0\), so the solutions is \(y(t)=c_{1} e^{r_{1} t}+c_{2} e^{r_{2} t}\). For Problems 30-35, discuss the long-term solution behaviors for the given r1, r2 combinations. Assume \(\beta \neq 0\). \(r=\alpha \pm \beta i\) ________________ Equation Transcription: Text Transcription: y''+by'+cy=0 y(t)=c_1e^r_1t+c_2e^r_2t Beta neq 0 r=alpha+-beta i

Long-Term Behavior of Solutions Suppose that r1 and r2 are the characteristic roots for \(y^{\prime \prime}+b y^{\prime}+c y=0\), so the solutions is \(y(t)=c_{1} e^{r_{1} t}+c_{2} e^{r_{2} t}\). For Problems 30-35, discuss the long-term solution behaviors for the given r1, r2 combinations. Assume \(\beta \neq 0\). \(r_{1}=0, r_{2}=0\) ________________ Equation Transcription: Text Transcription: y''+by'+cy=0 y(t)=c_1e^r_1t+c_2e^r_2t Beta neq 0 r_1=0, r_2=0

Long-Term Behavior of Solutions Suppose that r1 and r2 are the characteristic roots for \(y^{\prime \prime}+b y^{\prime}+c y=0\), so the solutions is \(y(t)=c_{1} e^{r_{1} t}+c_{2} e^{r_{2} t}\). For Problems 30-35, discuss the long-term solution behaviors for the given r1, r2 combinations. Assume \(\beta \neq 0\). \(r_{1}>0, r_{2}<0\) ________________ Equation Transcription: Text Transcription: y''+by'+cy=0 y(t)=c_1e^r_1t+c_2e^r_2t Beta neq 0 r_1>0, r_2<0

Long-Term Behavior of Solutions Suppose that r1 and r2 are the characteristic roots for \(y^{\prime \prime}+b y^{\prime}+c y=0\), so the solutions is \(y(t)=c_{1} e^{r_{1} t}+c_{2} e^{r_{2} t}\). For Problems 30-35, discuss the long-term solution behaviors for the given r1, r2 combinations. Assume \(\beta \neq 0\). \(r=\pm \beta i\) ________________ Equation Transcription: Text Transcription: y''+by'+cy=0 y(t)=c_1e^r_1t+c_2e^r_2t Beta neq 0 r=+- beta i

Linear Independence Verify that \(e^{\alpha t} \cos \beta t\) and \(e^{\alpha t} \sin \beta t\) are linearly independent on any interval ________________ Equation Transcription: Text Transcription: e^alpha t cos beta t e^alpha t sin beta t

Real Coefficients Suppose the roots of the characteristic equation for (1) are complex conjugates \(\alpha \pm i \beta\), which gives rise to the general solution \(y=k_{1} e^{(\alpha+i \beta) t}+k_{2} e^{(\alpha-i \beta) t}\) , where k1 and k2 are any constants (even complex). Show that in order for the solution y(t) to be real, k1 and k2 must be complex conjugates. ________________ Equation Transcription: Text Transcription: Alpha +- i beta y=k_1 e^(alpha+i beta)t + k_2 e^(alpha-i beta)t

Solving \(d^{n} y / d t^{n}=0\) (a) Solve the equation \(d^{4} y / d t^{4}=0\) by successive integration, getting \(d^{3} y / d t^{3}=k_{3}\) and \(d^{2} y / d t^{2}=k_{3} t+k_{2}\) to obtain y(t). (b) Determine the characteristic equation for(18) and use its roots to ?nd the general solution. Compare this solution with the solution you found in (a). (c) Generalize the process in (a) to solve \(d^{n} y / d t^{n}=0\). ________________ Equation Transcription: Text Transcription: d^n y/dt^n=0 d^4 y/dt^4=0 d^3y/dt^3=k_3 d^2y/dt^2=k_3t + k_2

Higher-Order DEs Find the solutions for the following higher-order equations. Remember that for each repetition of a root, a term with an additional factor of t must be included If a factorization of the characteristic equation f(t) = 0 is not obvious, look for a small integer that satisfies f(q) = 0. Divide the characteristic equation by r-q. \(y^{(5)}-4 y^{(4)}+4 y^{\prime \prime \prime}=0\) ________________ Equation Transcription: Text Transcription: y^(5)-4y^(4)+4y'''=0

Higher-Order DEs Find the solutions for the following higher-order equations. Remember that for each repetition of a root, a term with an additional factor of t must be included If a factorization of the characteristic equation f(t) = 0 is not obvious, look for a small integer that satisfies f(q) = 0. Divide the characteristic equation by r-q. \(y^{\prime \prime \prime}+4 y^{\prime \prime}-7 y^{\prime}-10 y=0\) ________________ Equation Transcription: Text Transcription: y'''+4y''-7y'-10y=0

Higher-Order DEs Find the solutions for the following higher-order equations. Remember that for each repetition of a root, a term with an additional factor of t must be included If a factorization of the characteristic equation f(t) = 0 is not obvious, look for a small integer that satisfies f(q) = 0. Divide the characteristic equation by r-q. \(y^{(5)}-y^{\prime}=0\) ________________ Equation Transcription: Text Transcription: y^(5)-y'=0

Higher-Order DEs Find the solutions for the following higher-order equations. Remember that for each repetition of a root, a term with an additional factor of t must be included If a factorization of the characteristic equation f(t) = 0 is not obvious, look for a small integer that satisfies f(q) = 0. Divide the characteristic equation by r-q. \(y^{\prime \prime \prime}-4 y^{\prime \prime}+5 y^{\prime}-2 y=0\) ________________ Equation Transcription: Text Transcription: y'''-4y''+5y'-2y=0

Higher-Order DEs Find the solutions for the following higher-order equations. Remember that for each repetition of a root, a term with an additional factor of t must be included If a factorization of the characteristic equation f(t) = 0 is not obvious, look for a small integer that satisfies f(q) = 0. Divide the characteristic equation by r-q. \(y^{\prime \prime \prime}+6 y^{\prime \prime}+12 y^{\prime}+8 y=0\) ________________ Equation Transcription: Text Transcription: y'''+6y''+12y'+8y=0

Higher-Order DEs Find the solutions for the following higher-order equations. Remember that for each repetition of a root, a term with an additional factor of t must be included If a factorization of the characteristic equation f(t) = 0 is not obvious, look for a small integer that satisfies f(q) = 0. Divide the characteristic equation by r-q. \(y^{(4)}-y=0\) ________________ Equation Transcription: Text Transcription: y^(4)-y=0

Linking Graphs For the sets of ty, ty', and yy' graphs in Problems 45 and 46, match the corresponding trajectories. They are numbered on the phase portrait, so you can use those same numbers to identi.fJ the curves in the component solution graphs. On each phase-plane trajectory. mark. the point where t = 0 and add arrowheads to show the direction of motion as t gets larger. Fig. 4.3.10

Linking Graphs For the sets of ty, ty', and yy' graphs in Problems 45 and 46, match the corresponding trajectories. They are numbered on the phase portrait, so you can use those same numbers to identi.fJ the curves in the component solution graphs. On each phase-plane trajectory. mark. the point where t = 0 and add arrowheads to show the direction of motion as t gets larger. Fig. 4.3.11

Changing the Damping Consider the mass-spring system \(\ddot{x}+b \dot{x}+x=0, \quad x(0)=4, \quad \dot{x}(0)=0 .\) For damping coefficient \(b=0,0.5, 1 , 2, 4\), the corresponding solutions are plotted together in Fig. 4.3. 1 2(a). Their phase-plane trajectories are plotted in Fig. 4.3. l 2(b ). Make a trace of both graphs and label each curve with the appropriate value of b. ________________ Equation Transcription: Text Transcription: Double dot x +b dot x + x = 0, x(0)=4, dot x(0)=0 b=0,0.5, 1 , 2, 4

Changing the Spring Consider the mass-spring system \(\ddot{x}+\dot{x}+k x=0, \quad x(0)=4, \quad \dot{x}(0)=0\). (a) For spring constant \(k =0.25, 0.5, I , 2, 4\), the corresponding solutions are plotted together in Fig. 4.3.I 3(a). Their phase-plane trajectories are plotted in Fig. 4.3. I 3(b). Make a trace of both graphs and label each curve with the appropriate value of k. (b) From your observations of the graphs in Fig. 4.3 .13, do the oscillations increase in frequency and amplitude as the spring constant is increased (i.e., as the spring becomes "stiffer")? Explain. ________________ Equation Transcription: Text Transcription: Double dot x + dot x + kx=0, x(0)=4, dot x (0)=0. k =0.25, 0.5, I , 2, 4

Changing the Mass A mass-spring system has a mass m attached in standard fashion with a damping factor \(b=0\) and a spring constant \(k = 16\). (a) Discuss how the value of m affects the motion. (b) How would the frequency of the motion be affected if the mass were doubled? (c) Discuss how much damping would be required for the critical damping if the mass were increased. ________________ Equation Transcription: Text Transcription: b=0 k = 16

Finding the Maximum (a) For the mass-spring system for which \(m = i, b = 2, k = 3\), and \(x(0)=1, \dot{x}(0)=0\), find the maximum displacement attained. (HINT: Differentiate the solution x(t) and set ?(t) = 0 to find the critical point.) (b) Do the same thing for \(m = I, b = 2, k = 10\), and \(x(0)=0, \dot{x}(0)=2\) (underdamped). (c) Do the same thing for \(m = I, b = 4, k = 4\), and \(x(0)=0, \dot{x}(0)=2\) (critically damped). ________________ Equation Transcription: x(0) = I, ?(0) = 0 ?(t) = 0 x(0) = 0, ?(0) = 2 x(0) = 0, ?(0) = 2 Text Transcription: m = i, b = 2, k = 3 x(0) = I, ?(0) = 0 ?(t) = 0 m = I, b = 2, k = 10 x(0) = 0, ?(0) = 2 m = I, b = 4, k = 4 x(0) = 0, ?(0) = 2

Oscillating Euler-Cauchy Euler-Cauchy equations were introduced in Sec. 4.2 Problem 60; Problems 51-54 consider Euler-Cauchy equations with non real characteristic roots. The solutions then have the final form \(y(t)=t^{\alpha}\left[c_{1} \cos (\beta \ln t)+c_{2} \sin (\beta \ln t)\right]\). Verify the solution (19). HINT: Use the relation \(t^{\alpha \pm \beta i}=e^{(\alpha \pm \beta i) \ln t}\). ________________ Equation Transcription: Text Transcription: y(t)=t^alpha[c_1 cos(beta ln t ) + c_2 sin(beta ln t )] t^alpha +- beta i=e^(alpha +- beta i) ln t

Oscillating Euler-Cauchy Euler-Cauchy equations were introduced in Sec. 4.2 Problem 60; Problems 51-54 consider Euler-Cauchy equations with non real characteristic roots. The solutions then have the final form \(y(t)=t^{\alpha}\left[c_{1} \cos (\beta \ln t)+c_{2} \sin (\beta \ln t)\right]\). Verify the solution (19). HINT: Use the relation \(\text { Solve } t 2 y^{\prime \prime}+2 t y^{\prime}+y=0\). ________________ Equation Transcription: Text Transcription: y(t)=t^alpha[c_1 cos(beta ln t ) + c_2 sin(beta ln t )] Solve t2y" + 2ty'+ y = 0

Oscillating Euler-Cauchy Euler-Cauchy equations were introduced in Sec. 4.2 Problem 60; Problems 51-54 consider Euler-Cauchy equations with non real characteristic roots. The solutions then have the final form \(y(t)=t^{\alpha}\left[c_{1} \cos (\beta \ln t)+c_{2} \sin (\beta \ln t)\right]\). Verify the solution (19). HINT: Use the relation \(\text { Solve } t 2 y^{\prime \prime}+3 t y^{\prime}+5 y=0\). ________________ Equation Transcription: Text Transcription: y(t)=t^alpha[c_1 cos(beta ln t ) + c_2 sin(beta ln t )] Solve t2y’’ + 3ty'+ 5y = 0

Oscillating Euler-Cauchy Euler-Cauchy equations were introduced in Sec. 4.2 Problem 60; Problems 51-54 consider Euler-Cauchy equations with non real characteristic roots. The solutions then have the final form \(y(t)=t^{\alpha}\left[c_{1} \cos (\beta \ln t)+c_{2} \sin (\beta \ln t)\right]\). Verify the solution (19). HINT: Use the relation \(\text { Solve } t 2 y^{\prime \prime}+17 t y^{\prime}+16 y=0\). ________________ Equation Transcription: Text Transcription: y(t)=t^alpha[c_1 cos(beta ln t ) + c_2 sin(beta ln t )] Solve t2y’’ + 17ty' + 16y = 0

Third-Order Euler-Cauchy Use the substitution y = t' for t > 0 to obtain the characteristic equation for the following third-order Euler-Cauchy equation. \(a t 3 y^{\prime \prime \prime}+b t 2 y^{\prime \prime}+c r y^{\prime}+d y=0 \text { for } t>0\). ________________ Equation Transcription: Text Transcription: at3y''' + bt2y'' +cry'+ dy = 0 for t > 0

Third-Order Euler-Cauchy Problems Use the results from Problem 55 to solve the specific Euler-Cauchy equations of Problems 56 and 57. \(r^{3} y^{\prime \prime \prime}+t^{2} y^{\prime \prime}-2 t y^{\prime}+2 y=0\) ________________ Equation Transcription: Text Transcription: r^3y''' + t^2y'' -2ty' + 2y = 0

Third-Order Euler-Cauchy Problems Use the results from Problem 55 to solve the specific Euler-Cauchy equations of Problems 56 and 57. \(r^{3} y^{\prime \prime \prime}+3 t^{2} y^{\prime \prime}+5 t y^{\prime}=0\) ________________ Equation Transcription: Text Transcription: r^3y''' + 3t^2y'' +5ty' = 0

Inverted Pendulum Since the general solution of the linearized pendulum equation \(\ddot{x}+(g / L) x=0\) is the class of sinusoidal oscillations, for small displacements, the pendulum oscillates back and forth about its equilibrium point. The equation \(\ddot{x}+(g / L) x=0\) describes the inverted pendulum. (See Fig. 4.3.14(b).) (a) Choosing g/L = 1, what is the motion of the inverted pendulum for \(x(0)=0 \text { and } \dot{x}(0)=1\)? (b) Are there initial conditions that will make the inverted pendulum approach \(x=\dot{x}=0 \text { as } t \rightarrow \infty\)? ________________ Equation Transcription: x(0)=0 and ?(0)=1 x = ? = 0 as Text Transcription: Double dot x + (g/L)x=0 x(0)=0 and dot x(0)=1 x=dot x = 0 as t rightarrow infinity

Pendulum and In"erted Pendulum For small displacements, where sin \(x \approx x\). the pendulum and invented pendulum of Fig. 4.3 . 1 4 are modeled (setting g/L = 1) by \(\ddot{x}+x=0\) (Linearized pendulum equation) \(\ddot{x}-x=0\) (Linearized inverted pendulum equation) Let us examine these models in the language of linear algebra. (a) Show that fundamental solutions of the pendulum equation are \(e^{i t}\) and \(e^{-i t}\) , while those of the inverted pendulum are \(e^{t}\) and \(e^{-t}\). (b) Show that fundamental solutions o f the pendulum equation are cos t and sin t, while those of the inverted pendulum equation are cosh t and sinh t (hyperbolic cosine and hyperbolic sine). (c) Are the solutions to both equations real? Explain. ________________ Equation Transcription: Text Transcription: x approx x double dot x + x = 0 double dot x - x = 0 e^it e^-it e^t e-t

Finding the Damped Oscillation Determine the constants for the damped oscillation \(x(t)=e^{-t}\left(c_{1} \cos t+c_{2} \sin t\right)\). subject to the initial conditions \(x(0)=1 \text { and } \dot{x}(0)=1\). Graph the solution. ________________ Equation Transcription: Text Transcription: x(t)=e^-t(c_1 cos t + c_2 sin t) x(0) = 1 and dot x (0) = 1

Extremes of Damped Oscillations Show that the maxima and minima of \(x(t)=e^{\alpha t}\left(c_{1} \cos \omega t+c_{2} \sin \omega t\right)\) for \(\alpha<0\) occur at equidistant values of t. Adjacent values during by \(\pi / \omega\). ________________ Equation Transcription: Text Transcription: x(t)=e^alpha t(c_1 cos omega t + c_2 sin omega t) alpha < 0 pi/omega

Underdamped Mass-Spring System Find and graph the motion of a damped mass-spring system with mass \(m=0.25 \text { slugs }\), spring constant \(k=4 \ \mathrm{lb} / \mathrm{ft}\), and damping constant \(b=1 \mathrm{lb} \ \mathrm{sec} / \mathrm{ft}\). The mass is initially pulled to the right, stretching the spring by 1 ft, and the released. ________________ Equation Transcription: Text Transcription: m=0.25 slugs k=4 lb/ft b=1 lb sec/ft

Damped Mass-Spring System The motion of a mass-spring system obeys \(\ddot{x}+b \dot{x}+64 x=0, \quad x(0)=1 \quad \dot{x}(0)=0 .\) Determine x(t) and sketch the motion for (a) \(b=10\); (b) \(b=16\); (c) \(b=20\). ________________ Equation Transcription: ; ; . Text Transcription: Double dot x + b dot x + 64 x =0, x(0)=1, dot x(0)=0.

LRC-Circuit I A series I.RC-circuit (Fig. 4.3.15) has a resistor of 8 ohms. an inductor of I henry and a capacitor of 0.04 farads. The initial charge on the capacitor is I coulomb, and there is initially no current in the circuit. Assume V(l) = 0 for t > 0. (a) Formulate the IVP for the charge across the capacitor. (b) Determine the charge across the capacitor for t > 0. (c) Find the current in the circuit for t > 0. (d) What are the long-term values of charge and current in the circuit?

LRC-Circuit II A series LRC-circuit has a resistor of I ohm, an inductor of 0.25 henries, and a capacitor of 0.25 farads. The initial charge on the capacitor is I coulomb, and there is initially no current in the circuit. Assume V(t) = 0 for t > 0. (a) Formulate the IVP for the charge across thC capacitor. (b) Determine the charge across the capacitor for t > 0. (c) Find the current in the circuit for t > 0. (d) What are the long-term values of charge and current in the circuit?

Computer Lab: Damped Free Vibrations Improve your understanding of damped oscillations by working through Lab 9 of the IDE software package, skipping over parts 1.7 and 2.5 on Energy.

Effects of Nonconstant Coeftldents In our study of the damped mass-spring system with mass m, spring constant k, and damping constant b, we have used as our model the second­ order linear differential \(m \ddot{x}+b \dot{x}+k x=0\) having constant coefficients. coefficients change with time, the analytic solutions we have found for constant coefficients do not work. Explain why. Then, for Problems 67-73, consider some DEs in which m, b, and k change with time. (a) Use your intuition and/or a computer to describe the motion of the system under these changing conditions. (b) Use a computer to draw a solution x(t) for t > 0, x(0) = 2, ?(0) = 0. HINT: When you need to avoid t = 0, try a trick like starting your plot at t = 0. 1. (c) Discuss what followed your intuition, what did not, and what further questions you might now ask. \(\ddot{x}+\frac{1}{t} x=0\) ________________ Equation Transcription: Text Transcription: M double dot x + b dot x + kx = 0 Double dot x + 1/t x = 0

Effects of Nonconstant Coeftldents In our study of the damped mass-spring system with mass m, spring constant k, and damping constant b, we have used as our model the second­ order linear differential \(m \ddot{x}+b \dot{x}+k x=0\) having constant coefficients. coefficients change with time, the analytic solutions we have found for constant coefficients do not work. Explain why. Then, for Problems 67-73, consider some DEs in which m, b, and k change with time. (a) Use your intuition and/or a computer to describe the motion of the system under these changing conditions. (b) Use a computer to draw a solution x(t) for t > 0, x(0) = 2, ?(0) = 0. HINT: When you need to avoid t = 0, try a trick like starting your plot at t = 0. 1. (c) Discuss what followed your intuition, what did not, and what further questions you might now ask. \(t \ddot{x}+x=0\) ________________ Equation Transcription: Text Transcription: M double dot x + b dot x + kx = 0 t Double dot x + x= 0

Effects of Nonconstant Coeftldents In our study of the damped mass-spring system with mass m, spring constant k, and damping constant b, we have used as our model the second­ order linear differential \(m \ddot{x}+b \dot{x}+k x=0\) having constant coefficients. coefficients change with time, the analytic solutions we have found for constant coefficients do not work. Explain why. Then, for Problems 67-73, consider some DEs in which m, b, and k change with time. (a) Use your intuition and/or a computer to describe the motion of the system under these changing conditions. (b) Use a computer to draw a solution x(t) for t > 0, x(0) = 2, ?(0) = 0. HINT: When you need to avoid t = 0, try a trick like starting your plot at t = 0. 1. (c) Discuss what followed your intuition, what did not, and what further questions you might now ask. \(\ddot{x}+(\sin t) \dot{x}+x=0\) ________________ Equation Transcription: Text Transcription: M double dot x + b dot x + kx = 0 Double dot x + (sin t)dot x + x = 0

Effects of Nonconstant Coeftldents In our study of the damped mass-spring system with mass m, spring constant k, and damping constant b, we have used as our model the second­ order linear differential \(m \ddot{x}+b \dot{x}+k x=0\) having constant coefficients. coefficients change with time, the analytic solutions we have found for constant coefficients do not work. Explain why. Then, for Problems 67-73, consider some DEs in which m, b, and k change with time. (a) Use your intuition and/or a computer to describe the motion of the system under these changing conditions. (b) Use a computer to draw a solution x(t) for t > 0, x(0) = 2, ?(0) = 0. HINT: When you need to avoid t = 0, try a trick like starting your plot at t = 0. 1. (c) Discuss what followed your intuition, what did not, and what further questions you might now ask. \(\ddot{x}+\frac{1}{t} \dot{x}+t x=0\) ________________ Equation Transcription: Text Transcription: M double dot x + b dot x + kx = 0 Double dot x + 1/t dot x + tx=0

Effects of Nonconstant Coeftldents In our study of the damped mass-spring system with mass m, spring constant k, and damping constant b, we have used as our model the second­ order linear differential \(m \ddot{x}+b \dot{x}+k x=0\) having constant coefficients. coefficients change with time, the analytic solutions we have found for constant coefficients do not work. Explain why. Then, for Problems 67-73, consider some DEs in which m, b, and k change with time. (a) Use your intuition and/or a computer to describe the motion of the system under these changing conditions. (b) Use a computer to draw a solution x(t) for t > 0, x(0) = 2, ?(0) = 0. HINT: When you need to avoid t = 0, try a trick like starting your plot at t = 0. 1. (c) Discuss what followed your intuition, what did not, and what further questions you might now ask. \(\ddot{x}+(\sin 2 t) x=0\) ________________ Equation Transcription: Text Transcription: M double dot x + b dot x + kx = 0 Double dot x + (sin 2t)x = 0

Boundary· Value Problems Two boundary conditions y(a1) =b1 and y(a2) = b1 can be used to specify the solution to what is now called a boundary-value problem, provided that the two conditions do not lead to a contradiction. Find all solutions of y’’ + y = 0 satisfying the boundary conditions in Problems 74-77. If the given boundary condition leads to a contradiction, state this fact explicitly and show that it is so. \(y(0)=0, y(\pi / 2)=0\) ________________ Equation Transcription: Text Transcription: y(0)=0, y(pi/2)=0

Boundary· Value Problems Two boundary conditions y(a1) =b1 and y(a2) = b1 can be used to specify the solution to what is now called a boundary-value problem, provided that the two conditions do not lead to a contradiction. Find all solutions of y’’ + y = 0 satisfying the boundary conditions in Problems 74-77. If the given boundary condition leads to a contradiction, state this fact explicitly and show that it is so. \(y(0)=0, y(\pi / 2)=1\) ________________ Equation Transcription: Text Transcription: y(0)=0, y(pi/2)=1

Boundary· Value Problems Two boundary conditions y(a1) =b1 and y(a2) = b1 can be used to specify the solution to what is now called a boundary-value problem, provided that the two conditions do not lead to a contradiction. Find all solutions of y’’ + y = 0 satisfying the boundary conditions in Problems 74-77. If the given boundary condition leads to a contradiction, state this fact explicitly and show that it is so. \(y(0)=1, y(\pi)=1\) ________________ Equation Transcription: Text Transcription: y(0)=1, y(pi)=1

Boundary· Value Problems Two boundary conditions y(a1) =b1 and y(a2) = b1 can be used to specify the solution to what is now called a boundary-value problem, provided that the two conditions do not lead to a contradiction. Find all solutions of y’’ + y = 0 satisfying the boundary conditions in Problems 74-77. If the given boundary condition leads to a contradiction, state this fact explicitly and show that it is so. \(y(0)=0, y(\pi / 2)=0\) ________________ Equation Transcription: Text Transcription: y(0)=0, y(pi/2)=0

Exact Second-Order Differential Equations The differential equation \(y^{\prime \prime}+p(t) y^{\prime}+q(t) y=0\) is called an exact second-order equation if it can be written in a form that can be integrated directly. An example is \(y^{\prime \prime}+[g(t) y]^{\prime}=0\). where g(t) is determined from p(t) and q(t). This DE can be integrated directly to get a first-order linear equation, which in turn can be integrated using the integrating factor method. For Problems 78-80, solve the given exact equations. \(y^{\prime \prime}+\frac{1}{t} y^{\prime}-\frac{1}{t} y=0\) (HINT: g(t)=1/t) ________________ Equation Transcription: Text Transcription: y'' + p(t)y' + q(t)y = 0 y'' + [g(t)y]' = 0 y''+1/ty'-1/t2y=0

Exact Second-Order Differential Equations The differential equation \(y^{\prime \prime}+p(t) y^{\prime}+q(t) y=0\) is called an exact second-order equation if it can be written in a form that can be integrated directly. An example is \(y^{\prime \prime}+[g(t) y]^{\prime}=0\). where g(t) is determined from p(t) and q(t). This DE can be integrated directly to get a first-order linear equation, which in turn can be integrated using the integrating factor method. For Problems 78-80, solve the given exact equations. \(y^{\prime \prime}+\frac{2}{t} y^{\prime}-\frac{2}{t} y=0\) ________________ Equation Transcription: Text Transcription: y'' + p(t)y' + q(t)y = 0 y'' + [g(t)y]' = 0 y''+2/ty'-2/t2y=0

Exact Second-Order Differential Equations The differential equation \(y^{\prime \prime}+p(t) y^{\prime}+q(t) y=0\) is called an exact second-order equation if it can be written in a form that can be integrated directly. An example is \(y^{\prime \prime}+[g(t) y]^{\prime}=0\). where g(t) is determined from p(t) and q(t). This DE can be integrated directly to get a first-order linear equation, which in turn can be integrated using the integrating factor method. For Problems 78-80, solve the given exact equations. \(\left(t^{2}-2 t\right) y^{\prime \prime}+4(t-1) y^{\prime}+2 y=0\) (HINT: Let g(t) = t2-2t) and show that the left-hand side equals (gy)’’.) ________________ Equation Transcription: Text Transcription: y'' + p(t)y' + q(t)y = 0 y'' + [g(t)y]' = 0 (t^2-2t)y''+4(t-1)y'+2y=0

Suggested Journal Entry The subsection entitled “Summarizing Solutions for Real and Complex Characteristic Roots" and Problems 30-35 on long-term behavior specify certain types of characteristic roots and tell how the solution evolves. Summarize these outcomes using the various behaviors as categories. That is. answer such questions as the following. (a) When do solutions tend to zero as \(t \rightarrow \infty\)? (b) When do solutions remain bounded but not tend to zero? (c) When do solutions oscillate in an unbounded manner? ________________ Equation Transcription: Text Transcription: t rightarrow infinity

Effects of Nonconstant Coeftldents In our study of the damped mass-spring system with mass m, spring constant k, and damping constant b, we have used as our model the second­ order linear differential \(m \ddot{x}+b \dot{x}+k x=0\) having constant coefficients. coefficients change with time, the analytic solutions we have found for constant coefficients do not work. Explain why. Then, for Problems 67-73, consider some DEs in which m, b, and k change with time. (a) Use your intuition and/or a computer to describe the motion of the system under these changing conditions. (b) Use a computer to draw a solution x(t) for t > 0, x(0) = 2, ?(0) = 0. HINT: When you need to avoid t = 0, try a trick like starting your plot at t = 0. 1. (c) Discuss what followed your intuition, what did not, and what further questions you might now ask. \(\ddot{x}+\frac{1}{t} \dot{x}+x=0\) ________________ Equation Transcription: Text Transcription: M double dot x + b dot x + kx = 0 Double dot x + 1/t x + x= 0

Effects of Nonconstant Coeftldents In our study of the damped mass-spring system with mass m, spring constant k, and damping constant b, we have used as our model the second­ order linear differential \(m \ddot{x}+b \dot{x}+k x=0\) having constant coefficients. coefficients change with time, the analytic solutions we have found for constant coefficients do not work. Explain why. Then, for Problems 67-73, consider some DEs in which m, b, and k change with time. (a) Use your intuition and/or a computer to describe the motion of the system under these changing conditions. (b) Use a computer to draw a solution x(t) for t > 0, x(0) = 2, ?(0) = 0. HINT: When you need to avoid t = 0, try a trick like starting your plot at t = 0. 1. (c) Discuss what followed your intuition, what did not, and what further questions you might now ask. \(\ddot{x}+\left(x^{2}-1\right) \dot{x}+x=0\) ________________ Equation Transcription: Text Transcription: M double dot x + b dot x + kx = 0 Double dot x + (x^2 - 1)dot x + x = 0