Use MATLAB to solve for and plot the unit-impulse response

Determine whether or not the following equations are linear or nonlinear, and state the reason for your answer. a. \(y \ddot{y}+5 \dot{y}+y=0\) b. \(\dot{y}+\sin y=0\) c. \(\dot{y}+\sqrt{y}=0\) d. \(\ddot{y}+5 t^{2} \dot{y}+3 y=0\) e. \(\ddot{y}+3 t^{2} \sin y=0\) f. \(\dot{y}+e^{t} y=0\)

Solve each of the following problems by direct integration: a. \(4 \dot{x}=3 t \quad x(0)=2\) b. \(5 \dot{x}=2 e^{-4 t} \quad x(0)=3\) c. \(3 \ddot{x}=5 t \quad x(0)=2 \quad \dot{x}(0)=7\) d. \(4 \ddot{x}=7 e^{-2 t} \quad x(0)=4 \quad \dot{x}(0)=2\) e. \(\ddot{x}=0 \quad x(0)=2 \quad \dot{x}(0)=5\)

Solve each of the following problems by separation of variables: a. \(\dot{x}+5 x^{2}=25 \quad x(0)=3\) b. \(\dot{x}-4 x^{2}=36 \quad x(0)=10\) c. \(x \dot{x}-5 x=25 \quad x(0)=4\) d. \(\dot{x}+2 e^{-4 t} x=0 \quad x(0)=5\)

Derive the Laplace transform of the ramp function x(t) = mt, whose slope is the constant m.

Extend the results of Problem 2.4 to obtain the Laplace transform of \(t^{2}\).

Obtain the Laplace transform of the following functions: a. \(x(t)=10+t^{2}\) b. \(x(t)=6 t e^{-5 t}+e^{-3 t}\) c. \(x(t)=t e^{-3 t} \sin 5 t\) d. \(x(t)=\left\{\begin{array}{ll} 0 & t<5 \\ t-5 & t>5 \end{array}\right.\)

Obtain the Laplace transform of the function shown in Figure P2.7.

Obtain the inverse Laplace transform f (t) for the following: a. \(\frac{6}{s^{2}+9}\) b. \(\frac{5}{s^{2}+4}+\frac{4 s}{s^{2}+4}\) c. \(\frac{6}{s^{2}+4 s+13}\) d. \(\frac{5}{s(s+3)}\) e. \(\frac{10}{(s+3)(s+7)}\) f. \(\frac{2 s+8}{(s+3)(s+7)}\)

Obtain the inverse Laplace transform f (t) for the following: a. \(\frac{5 s}{s^{2}+9}\) b. \(\frac{6}{s^{2}-9}\) c. \(\frac{45}{s(s+3)^{2}}\) d. \(\frac{2}{s\left(s^{2}+4 s+13\right)}\) e. \(\frac{20}{s\left(s^{2}+4\right)}\) f. \(\frac{20 s}{\left(s^{2}+4\right)^{2}}\)

Use the initial and final value theorems to determine x(0+) and \(x(\infty)\) for the following transforms: a. \(X(s)=\frac{5}{3 s+7}\) b. \(X(s)=\frac{10}{3 s^{2}+7 s+4}\)

Obtain the inverse Laplace transform x(t) for each of the following transforms: a. \(X(s)=\frac{6}{s(s+4)}\) b. \(X(s)=\frac{12 s+5}{s(s+3)}\) c. \(X(s)=\frac{4 s+7}{(s+2)(s+5)}\) d. \(X(s)=\frac{5}{s^{2}(2 s+8)}\) e. \(X(s)=\frac{3 s+2}{s^{2}(s+5)}\) f. \(X(s)=\frac{12 s+5}{(s+3)^{2}(s+7)}\)

Obtain the inverse Laplace transform x(t) for each of the following transforms: a. \(X(s)=\frac{7 s+2}{s^{2}+6 s+34}\) b. \(X(s)=\frac{4 s+3}{s\left(s^{2}+6 s+34\right)}\) c. \(X(s)=\frac{4 s+9}{\left(s^{2}+6 s+34\right)\left(s^{2}+4 s+20\right)}\) d. \(X(s)=\frac{5 s^{2}+3 s+7}{s^{3}+12 s^{2}+44 s+48}\)

Solve the following problems: a. \(5 \dot{x}=7 t \quad x(0)=3\) b. \(4 \dot{x}=3 e^{-5 t} \quad x(0)=4\) c. \(7 \ddot{x}=4 t \quad x(0)=3 \quad \dot{x}(0)=5\) d. \(3 \ddot{x}=8 e^{-4 t} \quad x(0)=3 \quad \dot{x}(0)=5\)

Solve the following problems: a. \(5 \dot{x}+7 x=0 \quad x(0)=4\) b. \(5 \dot{x}+7 x=15 \quad x(0)=0\) c. \(5 \dot{x}+7 x=15 \quad x(0)=4\) d. \(\dot{x}+7 x=4 t \quad x(0)=5\)

Solve the following problems: a. \(\ddot{x}+10 \dot{x}+21 x=0 \quad x(0)=4 \quad \dot{x}(0)=-3\) b. \(\ddot{x}+14 \dot{x}+49 x=0 \quad x(0)=1 \quad \dot{x}(0)=3\) c. \(\ddot{x}+14 \dot{x}+58 x=0 \quad x(0)=4 \quad \dot{x}(0)=-8\)

Solve the following problems: a. \(\ddot{x}+7 \dot{x}+10 x=20 \quad x(0)=5 \quad \dot{x}(0)=3\) b. \(5 \ddot{x}+20 \dot{x}+20 x=28 \quad x(0)=5 \quad \dot{x}(0)=8\) c. \(\ddot{x}+16 x=144 \quad x(0)=5 \quad \dot{x}(0)=12\) d. \(\ddot{x}+6 \dot{x}+34 x=68 \quad x(0)=5 \quad \dot{x}(0)=7\)

Solve the following problems: a. \(3 \ddot{x}+30 \dot{x}+63 x=5 \quad x(0)=\dot{x}(0)=0\) b. \(\ddot{x}+14 \dot{x}+49 x=98 \quad x(0)=\dot{x}(0)=0\) c. \(\ddot{x}+14 \dot{x}+58 x=174 \quad x(0)=\dot{x}(0)=0\)

Solve the following problems where \(x(0)=\dot{x}(0)=0\) a. \(\ddot{x}+8 \dot{x}+12 x=60\) b. \(\ddot{x}+12 \dot{x}+144 x=288\) c. \(\ddot{x}+49 x=147\) d. \(\ddot{x}+14 \dot{x}+85 x=170\)

Invert the following transforms: a. \(\frac{6}{s(s+5)}\) b. \(\frac{4}{(s+3)(s+8)}\) c. \(\frac{8 s+5}{2 s^{2}+20 s+48}\) d. \(\frac{4 s+13}{s^{2}+8 s+116}\)

Invert the following transforms: a. \(\frac{3 s+2}{s^{2}(s+10)}\) b. \(\frac{5}{(s+4)^{2}(s+1)}\) c. \(\frac{s^{2}+3 s+5}{s^{3}(s+2)}\) d. \(\frac{s^{3}+s+6}{s^{4}(s+2)}\)

Solve the following problems for x(t): a. \(5 \dot{x}+3 x=10+t^{2} \quad x(0)=2\) b. \(4 \dot{x}+7 x=6 t e^{-5 t}+e^{-3 t} \quad x(0)=5\) c. \(4 \ddot{x}+3 x=t e^{-3 t} \sin 5 t \quad x(0)=10 \quad \dot{x}(0)=-2\)

Compare the LCD method with equation (2.4.4) for obtaining the inverse Laplace transform of \(X(s)=\frac{7 s+4}{2 s^{2}+16 s+30}\)

Solve each of the following problems with the trial solution method. Identify the free, forced, transient, and steady-state responses. a. \(\ddot{x}+8 \dot{x}+15 x=30 \quad x(0)=10 \quad \dot{x}(0)=4\) b. \(\ddot{x}+10 \dot{x}+25 x=75 \quad x(0)=10 \quad \dot{x}(0)=4\) c. \(\ddot{x}+25 x=100 \quad x(0)=10 \quad \dot{x}(0)=4\) d. \(\ddot{x}+8 \dot{x}+65 x=130 \quad x(0)=10 \quad \dot{x}(0)=4\)

Determine whether the following models are stable, unstable, or neutrally stable: a. \(3 \dot{x}-5 x=12\) b. \(\ddot{x}-3 \dot{x}-10 x=50\) c. \(\ddot{x}-6 \dot{x}+34 x=68\) d. \(\dot{x}=3\) e. \(\ddot{x}+4 x=5\) f. \(\ddot{x}+5 \dot{x}=7\)

(a) Prove that the second-order system whose characteristic polynomial is \(m s^{2}+c s+k\) is stable if and only if m, c, and k have the same sign. (b) Derive the conditions for neutral stability.

For each of the following models, compute the time constant, if any. If no time constant exists, state why this is so. a. \(20 \dot{x}+4 x=0, \quad x(0)=6\) b. \(20 \dot{x}+5 x=15, \quad x(0)=3\) c. \(15 \dot{x}+5 x=0, \quad x(0)=-3\) d. \(8 \dot{x}-3 x=0, \quad x(0)=7\)

Obtain the steady-state response of each of the following models, and estimate how long it will take the response to reach steady state. a. \(13 \dot{x}+4 x=16 u_{s}(t), \quad x(0)=0\) b. \(13 \dot{x}+4 x=16 u_{s}(t), \quad x(0)=1\) c. \(15 \dot{x}-7 x=14 u_{s}(t), \quad x(0)=-2\)

Compare the responses of \(4 \dot{x}+x=\dot{g}(t)+g(t)\) and \(4 \dot{x}+x=g(t)\) if \(g(t)=5 u_{s}(t)\) and x(0?) = 0.

Obtain the response of the model \(3 \dot{x}+x=f(t)\), where f (t) is an impulse of strength 6 and x(0?) = 4.

If applicable, compute \(\zeta, \tau, \omega_{n}\), and \(\omega_{d}\) for the following characteristic polynomials. If not applicable, state the reason why. a. \(s^{2}+4 s+40=0\) b. \(s^{2}-2 s+24=0\) c. \(s^{2}+20 s+100=0\) d. s + 10 = 0

The characteristic equation of a certain system is \(s^{2}+10 d s+29 d^{2}=0\), where d is a constant. (a) For what values of d is the system stable? (b) Is there a value of d for which the free response will consist of decaying oscillations?

For each of the following equations, determine the transfer function X(s)/F(s) and compute the characteristic roots: a. \(5 \dot{x}+7 x=15 f(t)\) b. \(3 \ddot{x}+30 \dot{x}+63 x=5 f(t)\) c. \(\ddot{x}+10 \dot{x}+21 x=4 f(t)\) d. \(\ddot{x}+14 \dot{x}+49 x=7 f(t)\) e. \(\ddot{x}+14 \dot{x}+58 x=6 \dot{f}(t)+4 f(t)\) f. \(5 \dot{x}+7 x=4 \dot{f}(t)+15 f(t)\)

Obtain the transfer functions X(s)/F(s) and Y (s)/F(s) for the following model: \(3 \dot{x}=y \quad \dot{y}=f(t)-3 y-15 x\)

Obtain the transfer functions X(s)/F(s) and Y (s)/F(s) for the following model: \(\dot{x}=-2 x+5 y \quad \dot{y}=f(t)-6 y-4 x\)

a. Obtain the transfer functions X(s)/F(s) and Y (s)/F(s) for the following model: \(4 \dot{x}=y \quad \dot{y}=f(t)-3 y-12 x\) b. Compute \(\tau, \zeta, \omega_{n}\), and \(\omega_{d}\) for the model. c. If \(f(t)=u_{s}(t)\), will the responses x(t) and y(t) oscillate? If so, compute the radian oscillation frequency, and estimate how long it will take for the oscillations to disappear. d. Suppose that \(f(t)=u_{s}(t)\) and \(x(0)=\dot{x}(0)=0\). Obtain x(t) and y(t).

a. Obtain the transfer functions X(s)/F(s) and X(s)/G(s) for the following model: \(\dot{x}=-4 x+2 y+f(t) \quad \dot{y}=-9 y-5 x+g(t)\) b. Compute \(\tau, \zeta, \omega_{n}\), and \(\omega_{d}\) for the model. c. If f (t) = g(t) = 0, estimate how long it will take for the responses x(t) and y(t) to disappear.

Solve the following problems for x(t). Compare the values of x(0+) and x(0?). For parts (b) through (d), also compare the values of \(\dot{x}(0+)\) and \(\dot{x}(0-)\). a. \(7 \dot{x}+5 x=4 \delta(t) \quad x(0-)=3\) b. \(3 \ddot{x}+30 \dot{x}+63 x=5 \delta(t) \quad x(0-)=\dot{x}(0-)=0\) c. \(\ddot{x}+14 \dot{x}+49 x=3 \delta(t) \quad x(0-)=2 \quad \dot{x}(0-)=3\) d. \(\ddot{x}+14 \dot{x}+58 x=4 \delta(t) \quad x(0-)=4 \quad \dot{x}(0-)=7\)

Solve the following problems for x(t). The input g(t) is a unit-step function, \(g(t)=u_{s}(t)\). Compare the values of x(0+) and x(0?). For parts (c) and (d), also compare the values of \(\dot{x}(0+)\) and \(\dot{x}(0-)\). a. \(7 \dot{x}+5 x=4 \dot{g}(t) \quad x(0-)=3, \quad g(0-)=0\) b. \(7 \dot{x}+5 x=4 \dot{g}(t)+6 g(t) \quad x(0-)=3, \quad g(0-)=0\) c. \(3 \ddot{x}+30 \dot{x}+63 x=4 \dot{g}(t) \quad x(0-)=2 \quad \dot{x}(0-)=3, \quad g(0-)=0\) d. \(3 \ddot{x}+30 \dot{x}+63 x=4 \dot{g}(t)+6 g(t) \quad x(0-)=4 \quad \dot{x}(0-)=7, \quad g(0-)=0\)

Solve the following problem for x(t) and y(t): \(\begin{array}{c} 3 \dot{x}=y \quad x(0)=5 \\ \dot{y}=4 u_{s}(t)-3 y-15 x \quad y(0)=10 \end{array}\)

Solve the following problem for x(t) and y(t): \(\begin{array}{c} \dot{x}=-2 x+5 y \quad x(0)=5 \\ \dot{y}=-6 y-4 x+10 u_{s}(t) \quad y(0)=2 \end{array}\)

Determine the general form of the solution of the following equation, where the initial conditions y(0) and \(\dot{y}(0)\) have arbitrary values: \(\ddot{y}+y=e^{-t}\)

a. Use the Laplace transform to obtain the form of the solution of the following equation: \(\ddot{x}+4 x=3 t\) b. Obtain the solution to the equation in part (a) for the following conditions: x(0) = 10, x(5) = 30.

Obtain the inverse Laplace transform of \(X(s)=\frac{30}{\left(s^{2}+6 s+34\right)\left(s^{2}+36\right)}\)

Solve the following problem for x(t): \(\ddot{x}+12 \dot{x}+40 x=3 \sin 5 t \quad x(0)=\dot{x}(0)=0\)

Obtain the inverse transform in the form \(x(t)=A \sin (\omega t+\phi)\), where A > 0. \(X(s)=\frac{4 s+9}{s^{2}+25}\)

Use the Laplace transform to solve the following problem: \(\ddot{x}+6 \dot{x}+34 x=5 \sin 6 t \quad x(0)=0 \quad \dot{x}(0)=0\)

Express the oscillatory part of the solution of the following problem in the form of a sine function with a phase angle: \(\ddot{x}+12 \dot{x}+40 x=10 \quad x(0)=\dot{x}(0)=0\)

Find the steady-state difference between the input f (t) and the response x(t), if f (t) = 6t. \(\ddot{x}+8 \dot{x}+x=f(t) \quad x(0)=\dot{x}(0)=0\)

Invert the following transform: \(X(s)=\frac{1-e^{-3 s}}{s^{2}+6 s+8}\)

Obtain the Laplace transform of the function plotted in Figure P2.50.

Obtain the Laplace transform of the function plotted in Figure P2.51.

Obtain the Laplace transform of the function plotted in Figure P2.52.

Obtain the response x(t) of the following model, where the input P(t) is a rectangular pulse of height 3 and duration 5: \(4 \dot{x}+x=P(t) \quad x(0)=0\)

The Taylor series expansion for tan t is \(\tan t=t+\frac{t^{3}}{3}+\frac{2 t^{5}}{15}+\frac{17 t^{7}}{315}+\cdots \quad|t|<\frac{\pi}{2}\) Use the first three terms in the series to obtain an approximate closed-form solution of the following problem over the interval \(0 \leq t \leq 0.5\): \(\dot{x}+x=\tan t \quad x(0)=0\) Compare your answer at t = 0.5 with that obtained in Example 2.8.9, which was obtained by using two terms in the series.

Derive the initial value theorem: \(\lim _{s \rightarrow \infty} s X(s)=x(0+)\)

Derive the final value theorem: \(\lim _{s \rightarrow 0} s X(s)=\lim _{t \rightarrow \infty} x(t)\)

Derive the integral property of the Laplace transform: \(\mathcal{L}\left[\int_{0}^{t} x(t) d t\right]=\frac{X(s)}{s}+\left.\frac{1}{s} \int x(t) d t\right|_{t=0}\)

Use MATLAB to obtain the inverse transform of the following. If the denominator of the transform has complex roots, express x(t) in terms of a sine and a cosine. a. \(X(s)=\frac{8 s+5}{2 s^{2}+20 s+48}\) b. \(X(s)=\frac{4 s+13}{s^{2}+8 s+116}\) c. \(X(s)=\frac{3 s+2}{s^{2}(s+10)}\) d. \(X(s)=\frac{s^{3}+s+6}{s^{4}(s+2)}\) e. \(X(s)=\frac{4 s+3}{s\left(s^{2}+6 s+34\right)}\) f. \(X(s)=\frac{5 s^{2}+3 s+7}{s^{3}+12 s^{2}+44 s+48}\)

Use MATLAB to obtain the inverse transform of the following. If the denominator of the transform has complex roots, express x(t) in terms of a sine and a cosine. Hint: you will find it convenient to use the conv function to multiply two polynomials. a. \(X(s)=\frac{5}{(s+4)^{2}(s+1)}\) b. \(X(s)=\frac{4 s+9}{\left(s^{2}+6 s+34\right)\left(s^{2}+4 s+20\right)}\)

Use MATLAB to solve for and plot the unit-step response of the following models: a. \(3 \ddot{x}+21 \dot{x}+30 x=f(t)\) b. \(5 \ddot{x}+20 \dot{x}+65 x=f(t)\) c. \(4 \ddot{x}+32 \dot{x}+60 x=3 \dot{f}(t)+2 f(t)\)

Use MATLAB to solve for and plot the unit-impulse response of the following models: a. \(3 \ddot{x}+21 \dot{x}+30 x=f(t)\) b. \(5 \ddot{x}+20 \dot{x}+65 x=f(t)\)

Use MATLAB to solve for and plot the impulse response of the following model, where the strength of the impulse is 5: \(3 \ddot{x}+21 \dot{x}+30 x=f(t)\)

Use MATLAB to solve for and plot the step response of the following model, where the magnitude of the step input is 5: \(3 \ddot{x}+21 \dot{x}+30 x=f(t)\)

Use MATLAB to solve for and plot the response of the following models for \(0 \leq t \leq 1.5\), where the input is f (t) = 5t and the initial conditions are zero: a. \(3 \ddot{x}+21 \dot{x}+30 x=f(t)\) b. \(5 \ddot{x}+20 \dot{x}+65 x=f(t)\) c. \(4 \ddot{x}+32 \dot{x}+60 x=3 \dot{f}(t)+2 f(t)\)

Use MATLAB to solve for and plot the response of the following models for \(0 \leq t \leq 6\), where the input is f (t) = 6 cos 3t and the initial conditions are zero: a. \(3 \ddot{x}+21 \dot{x}+30 x=f(t)\) b. \(5 \ddot{x}+20 \dot{x}+65 x=f(t)\) c. \(4 \ddot{x}+32 \dot{x}+60 x=3 \dot{f}(t)+2 f(t)\)