Solution: 1–24 find the general solution of the given

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(\frac{d y}{d x}=5 y\) Text Transcription: dy/dx=5y

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(\frac{d y}{d x}+2 y=0\) Text Transcription: dy/dx+2y=0

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(\frac{d y}{d x}+y=e^{3 x}\) Text Transcription: dy/dx + y = e^3x

(a) The Fresnel sine integral is defined by \(S(x)=\int_{0}^{x} \sin \left(\pi t^{2} / 2\right) \ d t\). Express the solution y(x) of the initial-value problem \(y^{\prime}-\left(\sin x^{2}\right) y=0\), y(0) = 5, in terms of S(x). (b) Use a CAS to graph the solution curve for the IVP on \((-\infty, \infty)\). (c) It is known that \(S(x) \rightarrow \frac{1}{2}\) as \(x \rightarrow \infty\) and \(S(x) \rightarrow-\frac{1}{2}\) as \(x \rightarrow\ -\infty\) . What does the solution y(x) approach as \(x \rightarrow \infty\) ? As \(x \rightarrow\ -\infty\) Text Transcription: S(x)=int_0^x sin (pi t^2 / 2) dt y^prime - (sin x^2) y=0 (-infty, infty) S(x) rightarrow 1/2 x rightarrow infty S(x) rightarrow -1/2 x rightarrow -infty

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(3 \frac{d y}{d x}+12 y=4\) Text Transcription: 3 dy/dx + 12y = 4

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(y^{\prime}+3 x^{2} y=x^{2}\) Text Transcription: y^prime+3x^2 y=x^2

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(y^{\prime}+2 x y=x^{3}\) Text Transcription: y^prime+2xy=x^3

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(x^{2} y^{\prime}+x y=1\) Text Transcription: x^2 y^prime+xy=1

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(y^{\prime}=2 y+x^{2}+5\) Text Transcription: y^prime = 2y + x^2 + 5

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(x \frac{d y}{d x}-y=x^{2} \sin x\) Text Transcription: x dy/dx - y = x^2 sin x

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(x \frac{d y}{d x}+2 y=3\) Text Transcription: x dy/dx + 2y = 3

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(x \frac{d y}{d x}+4 y=x^{3}-x\) Text Transcription: x dy/dx + 4y = x^3 - x

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \((1+x) \frac{d y}{d x}-x y=x+x^{2}\) Text Transcription: (1+x) dy/dx - xy = x + x^2

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(x^{2} y^{\prime}+x(x+2) y=e^{x}\) Text Transcription: x^2 y^prime + x(x+2)y = e^x

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(x y^{\prime}+(1+x) y=e^{-x} \sin 2 x\) Text Transcription: xy^prime + (1+x)y = e^-x sin 2x

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(y \ d x-4\left(x+y^{6}\right) \ d y=0\) Text Transcription: y dx - 4(x+y^6)dy = 0

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(\cos ^{2} x \sin x \frac{d y}{d x}+\left(\cos ^{3} x\right) y=1\) Text Transcription: cos^2 xsin x dy/dx + (cos^3 x) y = 1

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(y \ d x=\left(y e^{y}-2 x\right) \ d y\) Text Transcription: y dx=(ye^y - 2x) dy

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \((x+1) \frac{d y}{d x}+(x+2) y=2 x e^{-x}\) Text Transcription: (x+1) dy/dx + (x+2) y = 2xe^-x

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \((x+2)^{2} \frac{d y}{d x}=5-8 y-4 x y\) Text Transcription: (x+2)^2 dy/dx = 5 - 8y - 4xy

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(\cos x \frac{d y}{d x}+(\sin x) y=1\) Text Transcription: cos x dy/dx+(sin x) y = 1

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(\frac{d P}{d t}+2 t P=P+4 t-2\) Text Transcription: dP/dt + 2tP = P + 4t - 2

In Problems 25–36 solve the given initial-value problem. Give the largest interval I over which the solution is defined \(\frac{d y}{d x}=x+5 y\), y(0)=3 Text Transcription: dy/dx=x+5y

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(\left(x^{2}-1\right) \frac{d y}{d x}+2 y=(x+1)^{2}\) Text Transcription: (x^2 - 1) dy/dx + 2y = (x+1)^2

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(x \frac{d y}{d x}+(3 x+1) y=e^{-3 x}\) Text Transcription: x dy/dx + (3x+1) y = e^-3x

In Problems 1–24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. \(\frac{d r}{d \theta}+r \sec \theta=\cos \theta\) Text Transcription: dr/d theta + rsec theta = cos theta

In Problems 25–36 solve the given initial-value problem. Give the largest interval I over which the solution is defined \(\frac{d y}{d x}=2 x-3 y\), \(y(0)=\frac{1}{3}\) Text Transcription: dy/dx=2x-3y y(0)=1/3

In Problems 25–36 solve the given initial-value problem. Give the largest interval I over which the solution is defined \(x y^{\prime}+y=e^{x}\), y(1)=2 Text Transcription: xy^prime + y = e^x

In Problems 25–36 solve the given initial-value problem. Give the largest interval I over which the solution is defined \(L \frac{d i}{d t}+R i=E\), \(i(0)=i_{0}\), L, R, E, \(i_{0}\) constants Text Transcription: L di/dt + Ri = E i(0) = i_0 i_0

In Problems 25–36 solve the given initial-value problem. Give the largest interval I over which the solution is defined \(y \frac{d x}{d y}-x=2 y^{2}\) y(1)=5 Text Transcription: y dx/dy - x = 2y^2

In Problems 25–36 solve the given initial-value problem. Give the largest interval I over which the solution is defined \(\frac{d T}{d t}=k\left(T-T_{m}\right)\), \(T(0)=T_{0}\), k, \(T_{m}\), \(T_{0}\) constants Text Transcription: dT/dt = k(T-T_m) T(0)=T_0 T_m T_0

In Problems 25–36 solve the given initial-value problem. Give the largest interval I over which the solution is defined \(x \frac{d y}{d x}+y=4 x+1\), y(1)=8 Text Transcription: x dy/dx + y = 4x+1

In Problems 25–36 solve the given initial-value problem. Give the largest interval I over which the solution is defined \((x+1) \frac{d y}{d x}+y=\ln x\), y(1)=10 Text Transcription: (x+1) dy/dx + y = ln x

In Problems 25–36 solve the given initial-value problem. Give the largest interval I over which the solution is defined \(y^{\prime}+4 x y=x^{3} e^{x^{2}}\), y(0)=-1 Text Transcription: y^prime + 4xy=x^3 e^x^2

In Problems 25–36 solve the given initial-value problem. Give the largest interval I over which the solution is defined \(x(x+1) \frac{d y}{d x}+x y=1\), y(e)=1 Text Transcription: x(x+1) dy/dx+xy=1

In Problems 25–36 solve the given initial-value problem. Give the largest interval I over which the solution is defined \(y^{\prime}-(\sin x) y=2 \sin x\), \(y(\pi/ 2)=1\) Text Transcription: y^prime - (sin x) y = 2 sin x y(pi/2)=1

In Problems 25–36 solve the given initial-value problem. Give the largest interval I over which the solution is defined \(y^{\prime}+(\tan x) y=\cos ^{2} x\), y(0)=-1 Text Transcription: y^prime+(tan x) y=cos^2 x

In Problems 37–40 proceed as in Example 6 to solve the given initial-value problem. Use a graphing utility to graph the continuous function y(x). \(\frac{d y}{d x}+2 y=f(x)\), y(0)=0, where \(f(x)=\left\{\begin{array}{lr}1, & 0 \leq x \leq 3 \\0, & x>3\end{array}\right.\) Text Transcription: dy/dx+2y=f(x) f(x)={1, 0 leq x leq 3 over 0, x>3

In Problems 37–40 proceed as in Example 6 to solve the given initial-value problem. Use a graphing utility to graph the continuous function y(x). \(\frac{d y}{d x}+y=f(x)\), y(0)=1, where \(f(x)=\left\{\begin{array}{lr}1, & 0 \leq x \leq 1 \\-1, & x>1\end{array}\right.\) Text Transcription: dy/dx+y=f(x) f(x)={1, 0 leq x leq 1 over -1, x>1

In Problems 37–40 proceed as in Example 6 to solve the given initial-value problem. Use a graphing utility to graph the continuous function y(x). \(\frac{d y}{d x}+2 x y=f(x)\), y(0)=2, where \(f(x)=\left\{\begin{array}{lr}x, & 0 \leq x <1 \\0, & x \geq 1\end{array}\right.\) Text Transcription: {dy/dx+2xy=f(x) f(x)={x, 0 leq x <1 over 0, x geq 1

In Problems 37–40 proceed as in Example 6 to solve the given initial-value problem. Use a graphing utility to graph the continuous function y(x). \(\left(1+x^{2}\right) \frac{d y}{d x}+2 x y=f(x)\), y(0)=0, where \(f(x)=\left\{\begin{array}{lr}x, & 0 \leq x <1 \\-x, & x \geq 1\end{array}\right.\) Text Transcription: (1+x^2) dy/dx + 2xy = f(x) f(x)={x, 0 leq x <1 over -x, x geq 1

Proceed in a manner analogous to Example 6 to solve the initial-value problem \(y^{\prime}+P(x) y=4 x\), y(0) = 3, where \(P(x)=\left\{\begin{array}{ll}2, & 0 \leq x \leq 1 \\-2 / x, & x>1\end{array}\right.\) Use a graphing utility to graph the continuous function y(x). Text Transcription: y^prime+P(x) y=4x P(x)={2, 0 leq x leq 1 over -2/x, x>1

Consider the initial-value problem \(y^{\prime}+e^{x} y=f(x)\), y(0) = 1. Express the solution of the IVP for x > 0 as a non elementary integral when f(x) = 1. What is the solution when f(x) = 0? When \(f(x)=e^{x}\) ? Text Transcription: y^prime+e^x y=f(x) f(x)=e^x

Express the solution of the initial-value problem \(y^{\prime}-2 x y=1\), y(1) = 1, in terms of erf(x). Text Transcription: y^prime-2xy=1

Reread the discussion following Example 2. Construct a linear first-order differential equation for which all nonconstant solutions approach the horizontal asymptote y = 4 as \(x \rightarrow \infty\). Text Transcription: x rightarrow infty

Reread Example 3 and then discuss, with reference to Theorem 1.2.1, the existence and uniqueness of a solution of the initial-value problem consisting of \(x y^{\prime}-4 y=x^{6} e^{x}\) and the given initial condition. (a) y(0) = 0 (b) \(y(0)=y_{0}\), \(y_{0}>0\) (c) \(y\left(x_{0}\right)=y_{0}\), \(x_{0}>0\), \(y_{0}>0\) Text Transcription: xy^prime-4y=x^6 e^x y(0)=y_0 y_0>0 y(x_0)=y_0 x_0>0 y_0>0

Reread Example 4 and then find the general solution of the differential equation on the interval (-3, 3).

Reread the discussion following Example 5. Construct a linear first-order differential equation for which all solutions are asymptotic to the line y = 3x - 5 as \(x \longrightarrow \infty\). Text Transcription: x longrightarrow infty

In determining the integrating factor (3), we did not use a constant of integration in the evaluation of \(\int P(x) \ d x\). Explain why using \(\int P(x) d x+c_{1}\) has no effect on the solution of (2). Text Transcription: int P(x) d x int P(x) dx+c_1

Reread Example 6 and then discuss why it is technically incorrect to say that the function in (9) is a “solution” of the IVP on the interval \([0, \infty)\). Text Transcription: [0, infty)

Heart Pacemaker A heart pacemaker consists of a switch, a battery of constant voltage \(E_{0}\), a capacitor with constant capacitance C, and the heart as a resistor with constant resistance R. When the switch is closed, the capacitor charges; when the switch is open, the capacitor discharges, sending an electrical stimulus to the heart. During the time the heart is being stimulated, the voltage E across the heart satisfies the linear differential equation \(\frac{d E}{d t}=-\frac{1}{R C} E\) Solve the DE, subject to \(E(4)=E_{0}\). Text Transcription: E_0 dE/dt=-1/RC E E(4)=E_0

Suppose P(x) is continuous on some interval I and a is a number in I. What can be said about the solution of the initial-value problem \(y^{\prime}+P(x) y=0\), y(a) = 0? Text Transcription: y^prime+P(x)y=0

(a) Construct a linear first-order differential equation of the form \(x y^{\prime}+a_{0}(x) y=g(x)\) for which \(y_{c}=c / x^{3}\) and \(y_{p}=x^{3}\) . Give an interval on which \(y=x^{3}+c / x^{3}\) is the general solution of the DE. (b) Give an initial condition \(y\left(x_{0}\right)=y_{0}\) for the DE found in part (a) so that the solution of the IVP is \(y=x^{3}-1 / x^{3}\) . Repeat if the solution is \(y=x^{3}+2 / x^{3}\) . Give an interval I of definition of each of these solutions. Graph the solution curves. Is there an initial-value problem whose solution is defined on \(-\infty, \infty\))? (c) Is each IVP found in part (b) unique? That is, can there be more than one IVP for which, say, \(y=x^{3}-1 / x^{3}\) , x in some interval I, is the solution? Text Transcription: xy^prime + a_0 (x)y=g(x) y_c=c/x^3 y_p=x^3 y=x^3 + c/x^3 y(x_0) = y_0 y=x^3 - 1/x^3 y=x^3 + 2/x^3 -infty, infty y=x^3 - 1/x^3

Radioactive Decay Series The following system of differential equations is encountered in the study of the decay of a special type of radioactive series of elements: \(\frac{d x}{d t}=-\lambda_{1} x\) \(\frac{d y}{d t}=\lambda_{1} x-\lambda_{2} y\) where \(\lambda_{1}\) and \(\lambda_{2}\) are constants. Discuss how to solve this system subject to \(x(0)=x_{0}\), \(y(0)=y_{0}\). Carry out your ideas. Text Transcription: dx/dt = -lambda_1 x dy/dt=lambda_1 x - lambda_2 y lambda_1 lambda_2 x(0)=x_0 y(0)=y_0

(a) Express the solution of the initial-value problem \(y^{\prime}-2 x y=-1\), \(y(0)=\sqrt{ } / 2\) in terms of erfc(x). (b) Use tables or a CAS to find the value of y(2). Use a CAS to graph the solution curve for the IVP on \((-\infty, \infty)\). Text Transcription: y^prime-2xy=-1 y(0)=sqrt / 2 (-infty, infty)

(a) The sine integral function is defined by \(\operatorname{Si}(x)=\int_{0}^{x}(\sin t / t) \ d t\), where the integrand is defined to be 1 at t = 0. Express the solution y(x) of the initial-value problem \(x^{3} y^{\prime}+2 x^{2} y=10 \sin x\), y(1) = 0 in terms of Si(x). (b) Use a CAS to graph the solution curve for the IVP for x > 0. (c) Use a CAS to find the value of the absolute maximum of the solution y(x) for x > 0. Text Transcription: Si(x) = int_0^x(sin t/t) dt x^3 y^prime + 2x^2 y = 10 sin x