What is the solution for y^{''}+2y=e^x+x^3 ?

The solution for y^{''}+2y=e^x+x^3 is y=c_{1}cos(sqrt(2)x)+c_{2}sin(sqrt(2)x)+1/3 e^x+(x^3-3x)/2

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y^{''}+2y=e^x+x^3

y^{^{''}} + 2 y = e^{x} + x^{3}

y = c_{1} cos (2 x) + c_{2} sin (2 x) + \frac{1}{3} e^{x} + \frac{x ^{3} - 3 x}{2}

Solution steps

y^{''} (x) + 2 y = e^{x} + x^{3}

Solve linear ODE:

y = c_{1} cos (2 x) + c_{2} sin (2 x) + \frac{1}{3} e^{x} + \frac{x ^{3} - 3 x}{2}

y = c_{1} cos (2 x) + c_{2} sin (2 x) + \frac{1}{3} e^{x} + \frac{x ^{3} - 3 x}{2}

f (x) = sin^{6} (x)

\int \frac{e ^{x}}{1 - 4 e ^{2 x}} d x

\frac{1.5}{( ( s + 1 ) )}

y^{^{'}} = e^{4 y} sin (5 x)

\frac{d}{d θ} (sin (3 θ))

What is the solution for y^{''}+2y=e^x+x^3 ?
The solution for y^{''}+2y=e^x+x^3 is y=c_{1}cos(sqrt(2)x)+c_{2}sin(sqrt(2)x)+1/3 e^x+(x^3-3x)/2