y=eaxsinbx,prove that (d2y/dx2)- 2a(dy/dx) + (a2+b2)y = 0 - Maths - Application of Derivatives

Question

y=eaxsinbx,prove that
(d2y/dx2)- 2a(dy/dx) +
(a2+b2)y = 0

Pronay · Accepted Answer

$y = e^{a x} \sin b x \frac{ⅆ y}{ⅆ x} = a e^{a x} \sin b x + e^{a x} \times b \cos b x$

$\frac{ⅆ^{2} y}{{ⅆ x}^{2}} = a^{2} e^{a x} \sin b x + a b e^{a x} \cos b x + a {b e}^{a x} \cos b x - b^{2} e^{a x} \sin b x$

$\frac{d^{2} y}{{d x}^{2}} - 2 a \frac{d y}{d x} + (a^{2} + b^{2}) y = (a^{2} e^{a x} \sin b x + a b e^{a x} \cos b x + a b e^{a x} \cos b x - b^{2} e^{a x} \sin b x) - 2 a (a e^{a x} \sin b x + e^{a x} b \cos b x) + (a^{2} + b^{2}) e^{a x} \sin b x = 0 (terms cancel each other)$