Question

If y=e^bx cos a x show that dy²/dx² -2b dv/dx + (a+b²)y=0​

Answer 1

Answer:

y=e^bx cos a x

or, dy/dx= be^bx cos a x-ae^bx sin a x

or, d^2y/dx^2 =b^2 e^bx cos a x- bae^bx sin a x-ab e^bx sin a x-a^2e^bx cos a x

Now,

dy²/dx² -2b dy/dx + (a+b²)y

=(b^2 e^bx cos a x- bae^bx sin a x-ab e^bx sin a x-a^2e^bx cos a x)-2b(be^bx cos a x-ae^bx sin a x)+(a+b^2)e^bx cos a x

=b^2 e^bx cos a x- bae^bx sin a x-ab e^bx sin a x-a^2e^bx cos a x-2b^2 e^bx cos a x+2bae^bx sin a x+(a+b^2)e^bx cos a x

=b^2 e^bx cos a x-a^2e^bx cos a x-2b^2 e^bx cos a x+(a+b^2)e^bx cos a x

=b^2 e^bx cos a x-a^2e^bx cos a x-2b^2 e^bx cos a x+ae^bx cos a x+b^2e^bx cos a x

=0

Hence,dy²/dx² -2b dv/dx + (a+b²)y=0,[proved]