Question

If y = eax cos bx, show that d2y / dx2 – 2 a dy/dx + ( a2 + b2 ) y = 0

Answer 1

Answer:

$y=e^{ax}*cos bx$

Differentiating once with respect to x,

$y'=e^{ax}*a*cos bx+e^{ax}*b*(-sin bx)\\\\y'=e^{ax}[a *cos bx- b* sin bx]$

Differentiating again with respect to x

$y''=a*e^{ax}*[a *cos bx- b* sin bx]+e^{ax}[a *b*(-sin bx)- b* b*cos bx]\\\\y"=a y'- a be^{ax}sin bx-b^2e^{ax}cos b x+a^2e^{ax}cos b x-a^2e^{ax}cos b x\\\\y"=a y'+ ae^{ax}[a *cos bx- b* sin bx]-(a^2+b^2)e^{ax}cos bx\\\\y"-2ay'+(a^2+b^2)y=0$

Hence Proved.

where, $y'=\frac{dy}{dx}, y"=\frac{d^2y}{dx^2}$