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Math Help - Need help with how to solve a certain type of differential equation

  1. #1
    Member iPod's Avatar
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    Need help with how to solve a certain type of differential equation

    I would really appreciate it if someone could show me the steps to how to solve such a differential equation.

    \dot{x}=32x-64y

    \dot{y}=-3x+48y

    x(0)=-64

    y(0)=4

    Thank you very much.
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  2. #2
    A Plied Mathematician
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    There are at least a couple of options:

    1. Use the eigenvalue/eigenvector diagonalization approach.
    2. Eliminate one variable to get a second-order DE.

    You can check those out at Chris's DE tutorial. Is a method specified for this problem, or can you use any method?
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  3. #3
    Member iPod's Avatar
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    its through the method of finding eigenvalues/vectors however i have no idea how to do so - if theres a tutorial online, id be very greatful
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    \displaystyle \frac{dx}{dt} = 32x - 64y
    \displaystyle \frac{dy}{dt} = -3x + 48y.


    \displaystyle \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}

    \displaystyle \frac{dy}{dx} = \frac{-3x + 48y}{32x - 64y}

    \displaystyle (32x - 64y)\,dy = (-3x + 48y)\,dx

    \displaystyle (32x - 64y)\,dy + (3x - 48y)\,dx = 0.


    Now you can probably find an integrating factor to turn this into an exact equation.
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  5. #5
    Member iPod's Avatar
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    :| you are amazing

    but ive forgotten how to find the integrating factor for exact equations, could you please remind me
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  7. #7
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    Quote Originally Posted by iPod View Post
    I would really appreciate it if someone could show me the steps to how to solve such a differential equation.

    \dot{x}=32x-64y

    \dot{y}=-3x+48y

    x(0)=-64

    y(0)=4

    Thank you very much.
    First write the system as a matrix.

    Let \mathbf{X}=\begin{pmatrix} x(t) \\ y(t) \end{pmatrix}

    And A=\begin{pmatrix} 32 & -64 \\ -3 & 48 \end{pmatrix}

    Now we have \dot{\mathbf{X}}=A\mathbf{X}

    Here is a link on how to solve a matrix system ODE. Scroll down a bit to find the computation of eigenvalues and eigenvectors section. They also have examples of this type of problem.
    Eigenvalues and Eigenvectors Technique
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