Exercises¶
1. The following ODE is of what type ?
\[
\begin{aligned}
\difftwo{u} - 2x^{2}u + \sin x = 0
\end{aligned}
\]
2. Solve the following differential equation
\[
\begin{aligned}
\diffone{y} &= \frac{1}{6e^{y} - 2x}
\end{aligned}
\]
3. Solve
\[
\begin{aligned}
\difftwo{y} = 1 + \roundbr{\diffone{y}}^{2}
\end{aligned}
\]
4. Solve
\[
\begin{aligned}
\difftwo{y} + \roundbr{1 + \frac{1}{y}}\roundbr{\diffone{y}}^{2} = 0
\end{aligned}
\]
5. Solve
\[
\begin{aligned}
x^{2}\difftwo{y} -3x\diffone{y} + 4y = 0
\end{aligned}
\]
6. Solve the IVP
\[
\begin{aligned}
\difftwo{y} + 3\diffone{y} + 2.25y = -10e^{-1.5x}
\end{aligned}
\]
7. Solve the following equation
\[
\begin{aligned}
x^{2}\difftwo{y} - x\diffone{y} + y = \ln x
\end{aligned}
\]
8. Solve the differential equation
\[
\begin{aligned}
\roundbr{1 + y^{2}}dx = \roundbr{\tan^{-1} y - x}dy
\end{aligned}
\]
9. Solve the equation
\[
\begin{aligned}
x^{3}\diffthree{y} - 3x^{2}\difftwo{y} + 6x\diffone{y} - 6y = x^{4}\ln x
\end{aligned}
\]
10. Find the radius of convergence
\[
\begin{aligned}
\sum_{m=0}^{\infty} \frac{x^{2m + 1}}{(2m + 1)!}
\end{aligned}
\]
11. Find the Laplace transform of
\[
\begin{aligned}
f(t) = \begin{cases}e^{t} &\mbox{$0 < t <\pi/2$}\newline 0 &\mbox{otherwise}\end{cases}
\end{aligned}
\]
12. Solve using Laplace Transform
\[
\begin{aligned}
\difftwo{y} + 6\diffone{y} + 8y &= e^{-3t} - e^{-5t}\newline
y(0) = \diffone{y}(0) &= 0
\end{aligned}
\]
13. Find
\[
\begin{aligned}
L^{-1}\roundbr{\ln\roundbr{1 + \frac{\omega^{2}}{s^{2}}}}
\end{aligned}
\]
14. Find
\[
\begin{aligned}
L\roundbr{\frac{1}{2}te^{-3t}}
\end{aligned}
\]
15. Find
\[
\begin{aligned}
L^{-1}\roundbr{\frac{s}{\roundbr{s^{2} - 9}^{2}}}
\end{aligned}
\]
16. Application of Laplace Transforms. Find
\[
\begin{aligned}
\int_{0}^{\infty} \frac{\sin t}{t} dt
\end{aligned}
\]