# Math Help - solving a system by substitution 3b

1. ## solving a system by substitution 3b

solve this system by sustituting $t=e^u$ and building a function $y(u)=x(e^u)$
$
t\frac{dx}{dt}=(\begin{array}{cc}
2 & 2\\
1 & 3\end{array})x
$

$
x(1)=(\begin{array}{c}
2\\
3\end{array})
$

how to do it?

2. Apply the chain rule to get: $\displaystyle \frac{dy}{du} = e^u \cdot \frac{dx}{dt} = t \cdot \frac{dx}{dt}$ ( $\frac{dx}{dt}$ evaluated at $e^u = t$ )

Then you have $\dot{y} = \begin{pmatrix}
2 & 2\\
1 & 3
\end{pmatrix} \cdot y$
now you could solve it, say, by computing the matrix exponential.

For the initial values think what knowing $x(1)$ would mean in terms of $y$, at the end, when you have $y$ you can undo the change of variables to get the solution for $t>0$.

3. you sat that y=x(u)t(u)=x(u)e^y
so the derivative is must have sum of two member
why you have one?