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\centerline{McGill University} \centerline{Math 325B: Differential
Equations} \centerline{Midterm Test} \vskip 1in
\begin{enumerate}
\item
Find the general solution of the differential equation
$$
x\frac{dy}{dx}=y+2xy^2.
$$
Find the solution $y$ with $y(1)=1$. What is the largest interval
on which this solution can be defined?
\bigskip

\item Show that the differential equation $M+Ny'=0$ has an
integrating factor $\mu$ which is a function of $xy$ if
$$
\frac{\frac{\partial N}{\partial x}-\frac{\partial M}{\partial
y}}{xM-yN}
$$
is a function of $xy$ and use this to find the general solution of
the differential equation
$$
xe^x\frac{dy}{dx}=xye^x-ye^x+y^2.
$$
\bigskip
\item Find the orthogonal trajectories of the family of curves
$y=Cx^4$.
\bigskip
\item Solve the initial value problem
$$
y'=x^2+xy,\quad y(0)=1
$$
using Picard iteration. If $y_n$ is the $n$-th Picard iteration,
give an estimate for $|y(x)-y_n(x)|$ on the interval $[-h,h]$.
\end{enumerate}
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