APM346 Self check: ODE

This is not a home assignment!

First order equations
  1. Find general solution $y'=x^2 y$ and solution satisfying Cauchy problem $y(0)=1$.
  2. Find general solution $y'+\tan (x)y=\cos(x)$ and solution satisfying Cauchy problem $y(0)=0$.

    Higher order equations
  3. Find general solution to $y''=y^2$ and solution satisfying Cauchy problem $y(0)=0$, $y'(0)=1$.

Linear equations of order $\ge 2$
  1. Find general solutions of equations \begin{align} &y''-4y=0,\label{eq-1}\\ &y''-4y'+4y=0,\label{eq-2}\\ &y''+4y=0,\label{eq-3}\\ &y''-5y'+6y=0,\label{eq-4}\\ &y''-4y'+5y=0,\label{eq-5}\\ &y^{(4)}-16 y=0.\label{eq-6} \end{align}
  2. Solve equations (\ref{eq-1})-(\ref{eq-5}) with the right-hand expressions $e^x$, $e^{2x}$, $\sin (x)$, $\sin (2x)$.
  3. Solve (\ref{eq-3}) with the right-hand expression $\tan(2x)$.
    Systems

There are resources from Fall 2012 and Spring 2013 classes to study and to check you knowledge of ODE. .