Programming Assignment Four

$\def\R{{\bf R}} \hangindent\parindent \item{6a.} Solve the initial value problem $$ y'=y\sin x\qquad\hbox{where}\qquad y(0)=1$$ exactly by multiplying through by the integrating factor $$ \mu=\exp\Big\{\int-\sin x \,dx\Big\}=e^{\displaystyle \cos x}$$ to obtain the equation $(\mu y)'=0$ and then integrating. Find $y(10)$ exactly. \medskip \item{6b.} Write a program that uses Euler's method to calculate approximate solutions $\tilde y$ for the above system. Use your program to find $\tilde y(10)$ using step sizes of $h=10/n$ where $n$ is $16, 32, 64, 128, \ldots, 1048576$. \medskip \item{6c.} Let ${\it E}=|y(10)-\tilde y(10)|$. Graph $\log E$ versus $\log h$ to verify the order of convergence for Euler's method numerically. \medskip \item{6d.} Use Runge-Kutta methods of order 2, 3 and 4 to calculate $\tilde y$. Graph $\log E$ versus $\log h$ to verify the order of convergence for each method. \medskip \item{6e.} Extra Credit: Repeat for the initial value problem $$ y'-y\sin x=1-x\sin x\qquad\hbox{where}\qquad y(0)=1.$$ Are the results the same?$

Last Updated: Thu Apr 19 17:53:44 PDT 2001