 # Exam 2 Review

Directions. The following are a few select review problems for Exam 2, but please note this review does not cover all sections on your Exam 2. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it.
1. Find the form of a particular solution for each of the following nonhomogeneous equations.
1. $$y''-2y'=(x^2-3x+1)e^{2x}+3x\cos(2x)+x^2$$
2. $$y''+6y'+5y=e^{-3t}+t\sin(2t)+t^2e^{-3t}\cos(2t)$$​

1. \begin{align}y_p=&x(Ax^2+Bx+C)e^{2x}+(Dx+E)\cos(2x)\\&+(Fx+G)\sin(2x)+x(Hx^2+Ix+J)\end{align}
2. \begin{align}y_p=&Ae^{-3t}+(Bt+C)\sin(2t)+(Dt+E)\cos(2t)\\&+t\left((Ft^2+Gt+H)\cos(2t)+(It^2+Jt+K)\sin(2t)\right)\end{align}

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2. Find the general solution of the equation.
$4y''-8y'+5y=e^x\tan^2\left(\frac{x}{2}\right)$​

$$y=e^x\left(c_1\cos(\frac{x}{2})+c_2\sin\left(\frac{x}{2}\right)+\sin\left(\frac{x}{2}\right)\ln\left|\sec\left(\frac{x}{2}\right)+\tan(\frac{x}{2})\right|-2\right)$$

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3. Use the definition of the Laplace transform to find the Laplace transform of the following function:
$f(t)=\left\{\begin{array}{ll}{t,} & {0 \leq t<1} \\ {2-t,} & {1 \leq t<2} \\ {0,} & {2 \leq t<\infty}\end{array}\right.$

$$F(s)=\dfrac{1}{s^2}-\dfrac{2}{s^2}e^{-s}+\dfrac{1}{s^2}e^{-2s}$$

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4. Find the Laplace transform of the following function (same as problem #3) using Heaviside's unit step function$f(t)=\left\{\begin{array}{ll}{t,} & {0 \leq t<1} \\ {2-t,} & {1 \leq t<2} \\ {0,} & {2 \leq t<\infty}\end{array}\right.$

$$F(s)=\dfrac{1}{s^2}-\dfrac{2}{s^2}e^{-s}+\dfrac{1}{s^2}e^{-2s}$$

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5. Find the inverse Laplace transform of the following functions
1. $$F(s) =\dfrac{2s^2+1}{s^4+3s^3-4s^2}$$
2. $$F(s)=\dfrac{s+3s e^{-5s}}{s^2-4s+3}$$

1. $$f(t)= \mathcal{L}^{-1}\{f\}-\dfrac{3}{16}-\dfrac{1}{4}t-\dfrac{33}{80}e^{-4t}+\dfrac{3}{5}e^t$$
2. $$f(t)= \mathcal{L}^{-1}\{f\}=-\dfrac{1}{2}e^t+\dfrac{3}{2}e^{3t}+u_5(t)\left(-\dfrac{3}{2}e^{t-5}+\dfrac{9}{2}e^{3(t-5)}\right)$$

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6. Solve the following initial value problem using the Laplace transform: $y''+y'+\dfrac{5}{4}y =\left\{\begin{array}{ll}{\sin t,} & {0 \leq t<\pi} \\ {0,} & {t \geq \pi}\end{array}\right. ; \quad y(0)=0, \quad y^{\prime}(0)=0$

$$y(t)=h(t)+u_{\pi}(t)h(t-\pi)$$, where $$h(t)=-\dfrac{16}{17}\cos(t)+\dfrac{4}{17}\sin(t)+\dfrac{1}{17}e^{-\frac{1}{2}t}[16\cos(t)+4\sin(t)]$$

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7. Use the definition of the Laplace transform to find the Laplace transform of the following function $f(t)=\delta(t-2)(4t^2-\cos(\pi t))$

$$F(s)=15e^{-2s}$$

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8. Find the Laplace transform of
1. $$f(t)= \displaystyle \int_{0}^t (t-\tau)e^{3\tau}\ d\tau$$
2. $$f(t)=\displaystyle \int_{0}^t (e^{\tau}\sin(t-\tau))\ d\tau$$​​

1. $$F(s)=\mathcal{L}\{f(t)\}=\dfrac{1}{s^2(s-3)}$$
2. $$F(s)=\mathcal{L}\{f(t)\}=\dfrac{1}{(s-1)(s^2+1)}$$

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