Find the general solution of the given second-order differential equation. y'' + 36y = 0.

Solution:

Given a differential equation y'' + 36y = 0.

We know that dy/dx = m

So the given equation becomes m² + 36 =0

m = ± 6i

The roots are the conjugate pair (α ± iβ) where α= 0 and β= 6

Let yc be the complementary solution then yc= c₁. e^αx.cos(βx) + c₂.e^αx.sin(βx)

Since α = 0,  β = 6, and “e” power zero is equal to 1, we have

⇒yc= c₁.cos(6x) + c₂.sin(6x)

The general solution is yc= c₁.cos(6x) + c₂.sin(6x)

Find the general solution of the given second-order differential equation. y'' + 36y = 0.

Summary:

The general solution of the given second-order differential equation. y'' + 36y = 0 is y= c₁.cos(6x) +c₂.sin(6x).