# Cramer's Rule application in differential equations

1. Jun 2, 2012

### Hiche

We have this system of equations:

$$\begin{cases} x'= -x + 2y & (1)\\ y' = -2x - y + e^{-t} & (2) \end{cases}$$

where $x(0) = 0 ; y(0) = 0$

We apply the Laplace transform on (1) and (2) and get:

$$(s + 1)X - 2Y = 0\\ 2X + (s + 1)Y = \frac{1}{s + 1}$$

We can use elimination here, but can we apply Cramer's Rule? We find the determinant $D = (s + 1)^2 + 4$, and to find $X_s$ and $Y_s$, we use a certain formula. Is this applicabale here?

2. Jun 4, 2012

### charbel

yes you can in fact Cramer's rule is only a shortcut to standard gaussian elimination