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Thread: s3.16.1 Find the general solution to the system of differential equations

  1. #1
    Super Member bigwave's Avatar
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    s3.16.1 Find the general solution to the system of differential equations

    Find the general solution to the system of differential equations

    $$\begin{cases}
    y'_1&=2y_1+y_2-y_3 \\y'_2&=3y_2+y_3\\y'_3&=3y_3
    \end{cases}$$

    ok I assume the next step is

    $$\begin{bmatrix}
    y'_1 \\y'_2 \\y'_3
    \end{bmatrix}=\begin{bmatrix}
    2 & 1 & -1 \\0 & 3 & 1 \\ 0 & 0 & 3
    \end{bmatrix}
    \begin{bmatrix}
    y_1 \\y_2 \\y_3
    \end{bmatrix}$$

    I had this posted on another forum but think I was not starting out right so got no reply's
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  2. #2
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    Re: s3.16.1 Find the general solution to the system of differential equations

    I would not use matrices at all. The third equation, $\displaystyle y_3'= 3y_3$, involves only $\displaystyle y_3$ and can be integrated directly: $\displaystyle y_3(t)= c_3e^{3t}$.

    The second equation is $\displaystyle y_2'= 3y_2+ y_3= 3y_2+ c_3e^{3t}$. That is a "non-homogeneous equation with constant equations". We can first solve the "associated homogeneous equation", $\displaystyle y_2'= 3y_2$. The general solution to that is $\displaystyle y= c_2e^{3t}$ just as before. We also seek a single solution to the entire equation. Normally, since the "non-homogeneous" part is $\displaystyle c_3e^{3t}$ we would try a multiple of that but that is already a solution to the homogeneous equation so we try $\displaystyle y= Ac_3te^{3t}$. Then $\displaystyle y'= Ac_3e^{3t}+ 3Ac_3te^{3t}$ so the equation becomes $\displaystyle Ac_3e^{3t}+ 3Ac_3te^{3t}= Ac_3te^{3t}+ c_3e^{3t}$. That reduces to $\displaystyle Ac_3e^{3t}= c_3e^{3t}$ so A= 1.
    $\displaystyle y_2= c_2e^{3t}+ c_3te^{3t}$.

    Now, the first equation is $\displaystyle y_1'= 2y_1+ y_2- y_3= 2y_1+ c_2e^{3t}+ 2c_3te^{3t}- c_3e^{3t}$. The associated homogeneous equation is $\displaystyle y_1'= 2y_1$ which has general solution $\displaystyle y= c_1e^{2t}$. To find a solution to the entire equation try $\displaystyle y= (Ax+ B)e^{3t}$ and find values for A and B that make that true.
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    Re: s3.16.1 Find the general solution to the system of differential equations

    Quote Originally Posted by HallsofIvy View Post
    I would not use matrices at all.
    But I suspect the point of the exercise is to know how to deal with repeated eigenvalues and rank deficient eigenvectors.
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