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\begin{document}

\begin{center}
\textbf{Math 2233.21569}\\[0pt]
SOLUTIONS TO FIRST EXAM \\[0pt]
February 23, 2021
\end{center}

1. \newline
(a) 2nd Order, Linear, PDE \newline
(b) 1st Order, Linear, ODE  \newline
(c) 2nd Order, Nonlnear, ODE \newline
(d) 3rd Order, Linear, ODE \newline
(e) 2nd Order, Nonlinear, PDE \newline

2. 
\begin{eqnarray*}
y_{1} &=&y_{0}+F\left( x_{0},y_{0}\right) \Delta x=1+\left(
x_{0}y_{0}+y_{0}\right) \left( 0.1\right) =1+\left( 0+1\right) \left(
0.1\right) =1.1 \\
y_{2} &=&y_{1}+\left( x_{1}y_{1}+y_{1}\right) \Delta x=1.1+\left(
(0.1)(1.1)+1.1\right) \left( 0.1\right) =1.221 \\
y_{3} &=&y_{2}+\left( x_{2}y_{2}+y_{2}\right) \Delta x=1.221+\left(
(0.2)(1.221)+1.221\right) \left( 0.1\right) =1.\,\allowbreak 367\,5
\end{eqnarray*}%
3.  
\begin{eqnarray*}
x+\sin \left( y\right) y^{\prime } &=&2 \\
M\left( x\right)  &=&x-2\quad ,\quad N\left( y\right) =\sin \left( y\right) 
\\
\cos \left( y\right)  &=&\frac{1}{2}x^{2}-2x-C \\
y\left( x\right)  &=&\cos ^{-1}\left( \frac{1}{2}x^{2}-2x-C\right) 
\end{eqnarray*}

4.  
\begin{eqnarray*}
xy^{\prime }-y &=&x\qquad ,\qquad y(1)=0 \\
y^{\prime }-\frac{1}{x}y &=&1\quad \Rightarrow \quad p\left( x\right) =-%
\frac{1}{x}\quad ,\quad g\left( x\right) =1 \\
\mu \left( x\right)  &=&\frac{1}{x} \\
y(x) &=&x\int \frac{1}{x}\left( 1\right) dx+Cx=x\ln \left\vert x\right\vert
+Cx
\end{eqnarray*}

5.%
\begin{equation*}
y+\left( x+1\right) \frac{dy}{dx}=x\qquad ,\qquad y\left( 0\right) =1
\end{equation*}%
\begin{eqnarray*}
\frac{\partial M}{\partial y} &=&1=\frac{\partial N}{\partial x} \\
\Phi \left( x,y\right)  &=&xy-\frac{1}{2}x^{2}+y \\
C &=&1 \\
y &=&\frac{1+\frac{1}{2}x^{2}}{x+1}
\end{eqnarray*}

6.%
\begin{equation}
x^{2}\frac{dy}{dx}=xy-y^{2}  \tag{*}
\end{equation}

Dividing both sides of the ODE by $x^{2}$ yields%
\begin{eqnarray*}
\frac{dy}{dx} &=&\frac{y}{x}-\left( \frac{y}{x}\right) ^{2} \\
x\frac{dz}{dx}+z &=&z-z^{2} \\
\ln \left\vert x\right\vert -\frac{1}{z} &=&C \\
y\left( x\right)  &=&\frac{x}{\ln \left\vert x\right\vert -C}
\end{eqnarray*}

\end{document}
