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Thread: Differential Equation System: Laplace

  1. #1
    Newbie cith's Avatar
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    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
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  2. #2
    MHF Contributor
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    Quote Originally Posted by cith View Post
    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.
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