Show that a given function satisfies a differential equation.

**zaboomafoo23** · February 26th 2010, 04:18 PM

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.

**Prove It** · February 26th 2010, 06:08 PM

Originally Posted by zaboomafoo23

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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