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.
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.
$\displaystyle y = e^{ax}\sin{bx}$
$\displaystyle y' = a\,e^{ax}\sin{bx} + b\,e^{ax}\cos{bx}$
$\displaystyle = e^{ax}(a\sin{bx} + b\cos{bx})$
$\displaystyle y'' = e^{ax}(a\cos{bx} - b\sin{bx}) + a\,e^{ax}(a\sin{bx} + b\cos{bx})$
$\displaystyle = e^{ax}(a\cos{bx} - b\sin{bx} + a^2\sin{bx} + ab\cos{bx})$
$\displaystyle = e^{ax}[(a + ab)\cos{bx} + (a^2 - b)\sin{bx}]$.
Now that you have $\displaystyle y$ and its derivatives, substitute them into the DE and see if you get what you are told you should get...