What is the solution for (d^2y)/(dx^2)+4y=x-sin(3x) ?

The solution for (d^2y)/(dx^2)+4y=x-sin(3x) is y=c_{1}cos(2x)+c_{2}sin(2x)+x/4+1/5 sin(3x)

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(d^2y)/(dx^2)+4y=x-sin(3x)

\frac{d ^{2} y}{d x ^{2}} + 4 y = x - sin (3 x)

y = c_{1} cos (2 x) + c_{2} sin (2 x) + \frac{x}{4} + \frac{1}{5} sin (3 x)

Solution steps

\frac{d ^{2} y}{d x ^{2}} + 4 y = x - sin (3 x)

Solve linear ODE:

y = c_{1} cos (2 x) + c_{2} sin (2 x) + \frac{x}{4} + \frac{1}{5} sin (3 x)

y = c_{1} cos (2 x) + c_{2} sin (2 x) + \frac{x}{4} + \frac{1}{5} sin (3 x)

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

y^{^{''}} - y^{^{'}} - 64 y = 54 cos (3 x)

y^{^{''}} - y^{^{'}} + 14 y = 7 + e^{\frac{x}{2}}

y^{^{'''}} - 5 y^{^{'}} + 6 y^{^{'}} = 9 + 5 sin (x)

(d + 2) (d - 1)^{2} y = e^{- 2} x + 2 sinh (x)

What is the solution for (d^2y)/(dx^2)+4y=x-sin(3x) ?
The solution for (d^2y)/(dx^2)+4y=x-sin(3x) is y=c_{1}cos(2x)+c_{2}sin(2x)+x/4+1/5 sin(3x)