Final Exam Review

$\newcount\qnum\newcount\qprt\qnum=0\qprt=0 \def\qn{\global\qprt=0\global\advance\qnum by1\bigskip \par\noindent\item{\bf\the\qnum.}} \def\qnn{\global\advance\qprt by1\bigskip \par\noindent\itemitem{\bf(\romannumeral\qprt)}} \parindent=1cm \qn Solve the initial value problem $$ ty'+2y=\sin(t),\qquad y(\pi/2)=1. $$$

$\qn Solve the initial value problem $$ y'=xy^3(1+x^2)^{-1/2},\qquad y(0)=1. $$$

$\qn State Theorem 2.4.2 from the book on the local existence and uniqueness for solutions of the initial value problem $$ {dy\over dt}=f(t,y),\qquad y(t_0)=y_0.$$$

$\qn Solve the equation $$ y\,dx+(2x-ye^y)dy=0.$$ Hint: Try an integrating factor $\mu=\mu(y)$.$

$\qn Solve the initial value problem $$ y''+y=te^t,\qquad y(0)=0,\qquad y'(0)=1.$$$

$\qn Solve the equation $$ y''+t(y')^2=0. $$$

$\qn State Theorem 3.7.1 from the book on the method of variation of parameters which tells how to find the general solution to the inhomogeneous equation $$ y''+p(t)y'+q(t)y=g(t) $$ given solutions $y_1$ and $y_2$ of the corresponding homogeneous equation.$

$\qn Determine a lower bound for the radius of convergence of the series solution about the point $x_0=4$ for the differential equation $$ (x^2-2x-3)y''+xy'+4y=0.$$$

$\qn Consider the second order linear differential equation $$ xy''+y'-y=0.$$ \qnn Show that $x_0=0$ is a regular singular point. \medskip \qnn Find the exponents at the singular point $x_0=0$. \medskip \qnn Find the first three nonzero terms in each of the two linearly independent solutions about $x_0=0$.$

$\qn Find the Laplace transform ${\cal L}\{f(t)\}$ of $f(t)=3t$.$

$\qn Define the convolution $(f*g)(t)$ of two functions $f(t)$ and $g(t)$.$

$\qn The matrix $$ A=\left[\matrix{2&1\cr 1&2}\right] $$ has eigenvectors $$\xi_1=\left[\matrix{-1\cr 1}\right] \qquad\hbox{and}\qquad \xi_2=\left[\matrix{1\cr 1}\right]$$ with corresponding eigenvalues $\lambda_1=1$ and $\lambda_2=3$. Solve the initial value problem $$x'=Ax,\qquad x(0)=\left[\matrix{1\cr 3}\right].$$$

$\qn The matrix $$ A=\left[\matrix{-3&2\cr -1&-1}\right] $$ has eigenvectors $$\xi_1=\left[\matrix{1\cr 1/2+i/2}\right] \qquad\hbox{and}\qquad \xi_2=\left[\matrix{1\cr 1/2-i/2}\right]$$ with corresponding eigenvalues $\lambda_1=-2+i$ and $\lambda_2=-2-i$. Find the solution to the initial value problem $$x'=Ax,\qquad x(0)=\left[\matrix{1\cr -2}\right].$$$

Last updated: Sun May 6 09:04:32 PDT 2001