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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. \(\begin{align}y_p=&Ae^{-3t}+(Bt+C)\sin(2t)+(Dt+E)\cos(2t)\\&+t\left((Ft^2+Gt+H)\cos(2t)+(It^2+Jt+K)\sin(2t)\right)\end{align}\)

      To see the full video page and find related videos, click the following link.
      MATH 308 WIR22A V66


  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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    MATH 308 WIR22A V67


  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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    MATH 308 WIR22A V54


  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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    MATH 308 WIR22A V55


  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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      MATH 308 WIR22A V56


  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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    MATH 308 WIR22A V57


  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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    MATH 308 WIR22A V68


  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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      MATH 308 WIR22A V69