Question

If y=eᵃˣ sinbx,prove y₂-2ay₁+(a²+b²)y=0

Answer 1

Answer:

$\bf{y_2-2a\:y_1+(a^2+b^2)y=0}$

Step-by-step explanation:

If y=eᵃˣ sinbx,prove y₂-2ay₁+(a²+b²)y=0

$y=e^{ax}\:sinbx$.......(1)

Using product rule differentiate with respect to x

$y_1=e^{ax}\:(cosbx)b+sinbx\:e^{ax}a$

$y_1=e^{ax}\:(cosbx)b+a\:y$......... (2) (using (1))

Using product rule differentiate with respect to x

$y_2=b[e^{ax}\:(-sinbx)b+cosbx\:e^{ax}a]+a\:y_1$ (using (2))

$y_2=(-b^2)y+(y_1-a\:y)a+a\:y_1$ (using (1))

$y_2=-b^2y+a\:y_1-a^2y+a\:y_1$

$y_2=-b^2y+2a\:y_1-a^2y$

$y_2=-(a^2+b^2)y+2a\:y_1$

$\implies\:\boxed{\bf{y_2-2a\:y_1+(a^2+b^2)y=0}}$