Solve the given initial value problem.

Question

Solve the given initial value problem.

y''+ 64y = 0 ; y(0)=1,   y'(0)=7

Answer 1

The given equation is

y''+ 64y = 0

Which is a second order homogeneous ODE. The characteristic equation corresponding to this ODE is

r^2 + 64 = 0

r = 8i, -8i

So, the solution is

y=C₁sin(8x)+C₂cos(8x)

which implies

y = C₁sin(8x)+C₂cos(8x)

y '= 8C₁cos(8x)-8C₂sin(8x)

Using the condition y'(0) = 1, we get C₁ = 1/8

Finally, using the condition y(0) = 1, we get C₂ = 1

Therefore, the solution is

y = cos(8x) + 1/8sin(8x)