What is the solution for y^{''}+4pi^2y=cos^3(2pix),y(0)=1 ?

The solution for y^{''}+4pi^2y=cos^3(2pix),y(0)=1 is

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y^{''}+4pi^2y=cos^3(2pix),y(0)=1

y^{^{''}} + 4 π^{2} y = cos^{3} (2 π x), y (0) = 1

y = (1 - \frac{1}{16 π ^{2}}) cos (2 π x) + c_{2} sin (2 π x) + \frac{1}{16 π ^{2}} cos^{5} (2 π x) + \frac{1}{4 π ^{2}} sin (2 π x) (cos^{3} (2 π x) sin (2 π x) + \frac{3}{8} (2 π x - \frac{1}{4} sin (8 π x)))

Solution steps

y^{''} (x) + 4 π^{2} y = cos^{3} (2 π x)

Solve linear ODE:

y = (1 - \frac{1}{16 π ^{2}}) cos (2 π x) + c_{2} sin (2 π x) + \frac{1}{16 π ^{2}} cos^{5} (2 π x) + \frac{1}{4 π ^{2}} sin (2 π x) (cos^{3} (2 π x) sin (2 π x) + \frac{3}{8} (2 π x - \frac{1}{4} sin (8 π x)))

y = (1 - \frac{1}{16 π ^{2}}) cos (2 π x) + c_{2} sin (2 π x) + \frac{1}{16 π ^{2}} cos^{5} (2 π x) + \frac{1}{4 π ^{2}} sin (2 π x) (cos^{3} (2 π x) sin (2 π x) + \frac{3}{8} (2 π x - \frac{1}{4} sin (8 π x)))

f (x) = 5 - 5 x^{2}, (5, - 120)

x \to 5 lim (x^{2} + 2 x - 4)

\int e^{13 x} d x

\frac{1}{2 s + 6}

\frac{d y}{d x} = π y + 12

What is the solution for y^{''}+4pi^2y=cos^3(2pix),y(0)=1 ?
The solution for y^{''}+4pi^2y=cos^3(2pix),y(0)=1 is