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Thread: the process of solving a differetial equation

  1. #1
    Member Jones's Avatar
    Apr 2006

    the process of solving a differetial equation

    Hi, im having some difficulties solving differential equations. It's not very well explained in my book.

    Take for example the equation $\displaystyle dy/dx -2y=4x$
    $\displaystyle y(0)=1$

    The first step is to check what kind of function you are dealing with. It's a first degree funktion where the general solution is $\displaystyle y=ax+b$
    or was it $\displaystyle C*e^{-ax}$

    Anyway if somebody please could explain the steps involved i'd be grateful

    As im lousy at math...

    It's fun when you understand what you're doing but can be hell if you don't.

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  2. #2
    Eater of Worlds
    galactus's Avatar
    Jul 2006
    Chaneysville, PA
    This DE is a 1st order linear.

    Find your integrating factor.

    $\displaystyle \frac{dy}{dx}-2y=4x$

    Integrating factor is $\displaystyle e^{\int{-2}dx}=e^{-2x}$

    $\displaystyle \frac{d}{dx}[e^{-2x}y]=e^{-2x}\cdot{4x}$

    Integrate both sides:

    $\displaystyle e^{-2x}y=-(2x+1)e^{-2x}+C$

    Solve for y:

    $\displaystyle y=-2x-1+Ce^{2x}$

    Use I.C. to find C:

    $\displaystyle 1=-2x-1+Ce^{2(0)}$


    $\displaystyle y=-2x-1+2e^{2x}$
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