# Differential Equation System: Laplace

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• May 20th 2009, 04:46 AM
cith
Differential Equation System: Laplace
I want to solve this system using the Laplace transform $\displaystyle (y_1=y_1(t),y_2=y_2(t))$

$\displaystyle y_1=-y_1+y_2 \hspace{0.5cm} y_1(0)=0$
$\displaystyle y_2=2y_1 \hspace{1.5cm} y_2(0)=1$

Probably not the hardest of problems, but I'm not even sure where to start :P
• May 20th 2009, 05:07 AM
Jester
Quote:

Originally Posted by cith
I want to solve this system using the Laplace transform $\displaystyle (y_1=y_1(t),y_2=y_2(t))$

$\displaystyle y_1=-y_1+y_2 \hspace{0.5cm} y_1(0)=0$
$\displaystyle y_2=2y_1 \hspace{1.5cm} y_2(0)=1$

Probably not the hardest of problems, but I'm not even sure where to start :P

First I'm assuming that you have differential equations

$\displaystyle y'_1=-y_1+y_2 \hspace{0.5cm} y_1(0)=0$
$\displaystyle y'_2=2y_1 \hspace{1.5cm} y_2(0)=1$

Let $\displaystyle L(y_1) = Y_1$ and $\displaystyle L(y_2) = Y_2$ so

$\displaystyle sY_1-0 = -Y_1 +Y_2$ and $\displaystyle sY_2 -1 = 2Y_2$. Solve for $\displaystyle Y_1$ and $\displaystyle Y_2$ and then take the Laplace inverses of each.