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Math 2214, Spring 2004, Final exam

1. Find the general solution of the differential equation

\begin{displaymath}y^{(5)}-y'=0.\end{displaymath}



2. Solve the system

\begin{displaymath}y_1'=-y_1+y_2,\end{displaymath}


\begin{displaymath}y_2'=-y_1-y_2,\end{displaymath}

with the initial condition

\begin{displaymath}y_1(0)=1,\quad y_2(0)=0.\end{displaymath}



3. For the matrix

\begin{displaymath}A=\pmatrix{2&1&1\cr 0&2&2\cr 0&0&3},\end{displaymath}

find the eigenvalues and their algebraic and geometric multiplicities. Identify the eigenvectors and, if applicable, generalized eigenvectors.

4. The system

\begin{displaymath}y'=\pmatrix{1&2\cr 3&4}y+\pmatrix{5\cr 6},\quad y(1)=\pmatrix{7\cr 8}\end{displaymath}

is solved using the Euler method with step size $h=0.01$. Find the resulting approximation for $y(1.01)$.

Answers:
1. $y=c_1+c_2e^t+c_3e^{-t}+c_4\cos t+c_5\sin t$.
2. $y_1=e^{-t}\cos t$, $y_2=-e^{-t}\sin t$.
3. The eigenvalues are 2 (algebraically double, geometrically simple) and 3 (simple). The eigenvector for 3 is (3,2,1), and the eigenvector for 2 is (1,0,0). A generalized eigenvector is given by (0,1,0).
4. $(7.28,8.59)$.

Michael Renardy 2004-09-21