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Math Help - Show that a given function satisfies a differential equation.

  1. #1
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    Show that a given function satisfies a differential equation.

    Show that the function y=e^(ax) sin(bx) satisfies the equation: y’’ — 2ay’ + (a2 + b2)y = 0 for any real constants a and b

    Any idea on how to solve this? Any help is appreciated.
    Last edited by mr fantastic; February 26th 2010 at 06:24 PM. Reason: Changed post title
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  2. #2
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    Quote Originally Posted by zaboomafoo23 View Post
    Show that the function y=e^(ax) sin(bx) satisfies the equation: y’’ — 2ay’ + (a2 + b2)y = 0 for any real constants a and b

    Any idea on how to solve this? Any help is appreciated.
    y = e^{ax}\sin{bx}


    y' = a\,e^{ax}\sin{bx} + b\,e^{ax}\cos{bx}

     = e^{ax}(a\sin{bx} + b\cos{bx})


    y'' = e^{ax}(a\cos{bx} - b\sin{bx}) + a\,e^{ax}(a\sin{bx} + b\cos{bx})

     = e^{ax}(a\cos{bx} - b\sin{bx} + a^2\sin{bx} + ab\cos{bx})

     = e^{ax}[(a + ab)\cos{bx} + (a^2 - b)\sin{bx}].



    Now that you have y and its derivatives, substitute them into the DE and see if you get what you are told you should get...
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