1. The following ODE is of what type ? \begin{align} \difftwo{u} - 2x^{2}u + \sin x = 0 \end{align}

    Solution

  2. Solve the following differential equation \begin{align} \diffone{y} &= \frac{1}{6e^{y} - 2x} \end{align}

    Solution

  3. Solve \begin{align} \difftwo{y} = 1 + \roundbr{\diffone{y}}^{2} \end{align}

    Solution

  4. Solve \begin{align} \difftwo{y} + \roundbr{1 + \frac{1}{y}}\roundbr{\diffone{y}}^{2} = 0 \end{align}

    Solution

  5. Solve \begin{align} x^{2}\difftwo{y} -3x\diffone{y} + 4y = 0 \end{align}

    Solution

  6. Solve the IVP \begin{align} \difftwo{y} + 3\diffone{y} + 2.25y = -10e^{-1.5x} \end{align}

    Solution

  7. Solve the following equation \begin{align} x^{2}\difftwo{y} - x\diffone{y} + y = \ln x \end{align}

    Solution

  8. Solve the differential equation \begin{align} \roundbr{1 + y^{2}}dx = \roundbr{\tan^{-1} y - x}dy \end{align}

    Solution

  9. Solve the equation \begin{align} x^{3}\diffthree{y} - 3x^{2}\difftwo{y} + 6x\diffone{y} - 6y = x^{4}\ln x \end{align}

    Solution

  10. Find the radius of convergence \begin{align} \sum_{m=0}^{\infty} \frac{x^{2m + 1}}{(2m + 1)!} \end{align}

    Solution

  11. Find the Laplace transform of \begin{align} f(t) = \begin{cases}e^{t} &\mbox{$0 < t <\pi/2$}\newline 0 &\mbox{otherwise}\end{cases} \end{align}

    Solution

  12. Solve using Laplace Transform \begin{align} \difftwo{y} + 6\diffone{y} + 8y &= e^{-3t} - e^{-5t}\newline y(0) = \diffone{y}(0) &= 0 \end{align}

    Solution

  13. Find \begin{align} L^{-1}\roundbr{\ln\roundbr{1 + \frac{\omega^{2}}{s^{2}}}} \end{align}

    Solution

  14. Find \begin{align} L\roundbr{\frac{1}{2}te^{-3t}} \end{align}

    Solution

  15. Find \begin{align} L^{-1}\roundbr{\frac{s}{\roundbr{s^{2} - 9}^{2}}} \end{align}

    Solution

  16. Application of Laplace Transforms. Find \begin{align} \int_{0}^{\infty} \frac{\sin t}{t} dt \end{align}

    Solution