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{\Large MATH 411, Spring 2001} \\[2mm]
{\LARGE Homework 4} \\[2mm]
{\large Due: 5, April, 2001}\\[2mm]
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R. Redheffer, {\it Differential Equations}:

[13.1] 3(i,m);

[13.2] 9;

[15.3] 6(b) or 6(a) (the one which your prefer most);

*) Construct fundamental matrix for each problem and find the solution 
satisfying specified initial data  
$$
{\rm a)} 
\qquad
\left\{
\begin{array}{rcl}
\frac{d x}{d t} & = & 2x+y, \\[2mm]
 \frac{d y}{d t} & = & x-3y, 
\end{array}
\right.\qquad
x(0)=y(0)=0,
$$
$$
{\rm b)} 
\qquad
\left\{
\begin{array}{rcl}
\frac{d x}{d t}  & = & 2x-y+z, \\[2mm]
 \frac{d y}{d t} & = & x+z,\\[2mm] 
\frac{d z}{d t}  & = & y-2z-3x, \\[2mm]
\end{array}
\right.\qquad
x(0)=0,\ y(0)=0,\ z(0)=1.
$$


**) Find the general solution for the inhomogeneous linear system by method of
variation of parameter. 
$$
{\rm a)} 
\qquad
\left\{
\begin{array}{rcl}
\frac{d x}{d t}+2x-y  & = & -{\rm e}^{2t}, \\[2mm]
\frac{d y}{d t}+3x-2y & = & 6{\rm e}^{2t}, 
\end{array}
\right.
$$
$$
{\rm b)} 
\qquad
\left\{
\begin{array}{rcl}
\frac{d x}{d t} & = & y, \\[2mm]
\frac{d y}{d t} & = & -x+\frac{1}{\cos t}, 
\end{array}
\right.
$$


All problems are estimated by 20 points.
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