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\begin{center}
\textbf{Math 2233.21570} \\%[0pt]
Final Exam \\%[0pt]
2:00pm -- 4:00pm \\%[0pt]
May 4, 2021
\end{center}

\begin{flushright}
Name:\underline{\hspace{2.5 in}}
\end{flushright}

1. (15 pts) Solve the following initial value problem
\[
xy^{\prime}-3y=x^{2}\qquad,\qquad y(1)=3\quad.
\]


\vspace{4in}

2. (15 pts) Show that the following equation is exact and then find the
explicit solution.
\[
\left(  3x+1\right)  \frac{dy}{dx}=2x-3y
\]


\newpage

3. (15 pts) Given that $y_{1}\left(  x\right)  =e^{2x}$ is one solution of
\begin{equation}
y^{\prime\prime}-4y^{\prime}+4y=0\quad, \tag{*}%
\end{equation}
use Reduction of Order (explicitly)\ to find the general solution of (*).

\vspace{3.5in}

4. (20 pts) Solve the following initial value problem
\[
y^{^{\prime\prime}}-2y^{\prime}-3y=e^{2x}\quad,\quad y\left(  1\right)
=2\quad,\quad y^{\prime}\left(  1\right)  =1
\]
(Hint: the corresponding homogeneous equation is a constant coefficients type ODE)

\newpage

5. (15 pts) Find the function $f\left(  x\right)  $ whose Laplace transform is
$\displaystyle F\left(  s\right)  =\frac{s+1}{\left(  s+2\right)  \left(
s+3\right)  }.$

\vspace{3.5in}

6. (15 pts) Solve the following initial value problems using the Laplace
transform method.%
\begin{align*}
y^{\prime\prime}+3y^{\prime}+2y &  =0\\
y\left(  0\right)   &  =2\\
y^{\prime}\left(  0\right)   &  =-4
\end{align*}


\newpage

7. (15 pts) Reduce the following expression to a single power series:%
\[
x\sum_{n=0}^{\infty}a_{n}\left(  x+2\right)  ^{n}+\sum_{n=0}^{\infty}%
a_{n}\left(  x+2\right)  ^{n}%
\]


\vspace{4in}

8. (15 pts) Find the recursion relations for a power series solution about
$x_{o}=0$ for the following differential equation.
\[
y^{\prime\prime}-\left(  x+1\right)  y^{\prime}-y=0
\]


\newpage

9. (10 pts) \textbf{Given} that the recursion relations for $y^{\prime\prime
}-2xy^{\prime}-2y=0$ about $x_{o}=1$ are
\[
a_{n+2}=\frac{2a_{n}+2a_{n+1}}{n+2}\qquad,\qquad n=0,1,2,3,\ldots
\]
Write down the first 4 terms of the power series solution satisfying $y(1)=1$,
$y^{\prime}(1)=2$. (I.e., find the solution up to order $\left(  x-1\right)
^{3}$.)

\vspace{4in}

10. Consider the following differential equation:
\[
\left(  1-x^{2}\right)  y^{\prime\prime}+xy^{\prime}+\frac{1}{(x-2)}y=0
\]


(a) (5 pts) Identify the singular points of this differential equation.

(b) (10 pts) For what range of $x$ is a power series solution of the form
$\sum_{n=0}^{\infty}a_{n}\left(  x+2\right)  ^{n}$ guaranteed to converge?

\newpage

\begin{center}
~ \vspace{1in} {\Huge \textbf{Table of Laplace Transforms} }
\end{center}

%

\[
\mathcal{L}\left[  \alpha f(t) + \beta g(t) \right]  = \alpha\mathcal{L}%
\left[  f(t) \right]  + \beta\mathcal{L}\left[  g(t) \right]  \quad\text{ if }
\alpha, \beta\text{ are constants}
\]
\[
\mathcal{L}\left[  \frac{df}{dt}\right]  =s\mathcal{L}\left[  f\right]
+f\left(  0\right)
\]%
\[
\mathcal{L}\left[  \frac{d^{2}f}{dx^{2}}\right]  =s^{2}\mathcal{L}\left[
f\right]  +sf\left(  0\right)  +\frac{df}{dt}\left(  0\right)
\]
\[
\mathcal{L}\left[  t^{n}\right]  =\frac{n!}{s^{n+1}}%
\]%
\[
\mathcal{L}\left[  e^{at}\right]  =\frac{1}{s-a}%
\]%
\[
\mathcal{L}\left[  \sin(at)\right]  =\frac{a}{s^{2}+a^{2}}%
\]%
\[
\mathcal{L}\left[  \cos(at)\right]  =\frac{s}{s^{2}+a^{2}}%
\]%
\[
\mathcal{L}\left[  \sinh\left(  at\right)  \right]  =\frac{a}{s^{2}-a^{2}}%
\]%
\[
\mathcal{L}\left[  \cosh\left(  at\right)  \right]  =\frac{s}{s^{2}-a^{2}}%
\]%
\[
\mathcal{L}\left[  e^{at}\sin\left(  bt\right)  \right]  =\frac{b}{\left(
s-a\right)  ^{2}+b^{2}}%
\]%
\[
\mathcal{L}\left[  e^{at}\cos\left(  bt\right)  \right]  =\frac{s-a}{\left(
s-a\right)  ^{2}+b^{2}}%
\]%
\[
\mathcal{L}\left[  t^{n}e^{at}\right]  =\frac{n!}{\left(  s-a\right)  ^{n+1}}%
\]%
\[
\mathcal{L}\left[  e^{at}\sinh\left(  bt\right)  \right]  =\frac{b}{\left(
s-a\right)  ^{2}-b^{2}}%
\]%
\[
\mathcal{L}\left[  e^{at}\cosh\left(  bt\right)  \right]  =\frac{s-a}{\left(
s-a\right)  ^{2}-b^{2}}%
\]%
\[
\mathcal{L}\left[  u\left(  t-a\right)  f\left(  t\right)  \right]
=e^{-at}\mathcal{L}\left[  f\left(  t+a\right)  \right]
\]%
\[
\mathcal{L}^{-1}\left[  e^{-as}\mathcal{L}\left[  f\right]  \right]  =u\left(
t-a\right)  f\left(  t-a\right)
\]



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