Find a particular solution to the nonhomogeneous differential equation.

Question

Find a particular solution to the nonhomogeneous differential equation:

y" + 16y = sec(4x)

I know that the most general solution to the associated homogeneous differential equation is C₁sin(4x) + C₂cos(4x).

Answer 1

y₁ = cos4x, y₂ = sin4xx; vronscian W = detI(cos4x sin4x), (- 4sin4x 4cos4x) = 4cos²4x + 4sin²4x = 4.

now, particular solution of given nonhomogenius by variation of parameters formula:

y_P = - cos4x∫sin4xsec4x/4dx + sin4x∫cos4xsec4x/4dx = 1/16cos4xlnIcos4xI + 1/4xsin4x

So, general solution of given ODE: y = C₁cos4x + C₂sin4x + 1/16cos4xlnIcos4xI + 1/4xsin4x