What is the solution for (d^2y)/(dx^2)+(dy)/(dx)-y=xsinh(x) ?

The solution for (d^2y)/(dx^2)+(dy)/(dx)-y=xsinh(x) is

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(d^2y)/(dx^2)+(dy)/(dx)-y=xsinh(x)

\frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} - y = x sinh (x)

y = c_{1} e^{\frac{( - 1 + 5 ) x}{2}} + c_{2} e^{\frac{( - 1 - 5 ) x}{2}} + e^{\frac{- x + 5 x}{2}} \cdot \int \frac{5 e ^{\frac{x - 5 x}{2}} x sinh ( x )}{5} d x + e^{\frac{- x - 5 x}{2}} \cdot \int - \frac{5 e ^{\frac{x + 5 x}{2}} x sinh ( x )}{5} d x

Solution steps

\frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} - y = x sinh (x)

Solve linear ODE:

y = c_{1} e^{\frac{( - 1 + 5 ) x}{2}} + c_{2} e^{\frac{( - 1 - 5 ) x}{2}} + e^{\frac{- x + 5 x}{2}} \cdot \int \frac{5 e ^{\frac{x - 5 x}{2}} x sinh ( x )}{5} d x + e^{\frac{- x - 5 x}{2}} \cdot \int - \frac{5 e ^{\frac{x + 5 x}{2}} x sinh ( x )}{5} d x

y = c_{1} e^{\frac{( - 1 + 5 ) x}{2}} + c_{2} e^{\frac{( - 1 - 5 ) x}{2}} + e^{\frac{- x + 5 x}{2}} \cdot \int \frac{5 e ^{\frac{x - 5 x}{2}} x sinh ( x )}{5} d x + e^{\frac{- x - 5 x}{2}} \cdot \int - \frac{5 e ^{\frac{x + 5 x}{2}} x sinh ( x )}{5} d x

y^{^{''}} + y = sin (x)

y^{^{'''}} + 12 y^{^{''}} + 48 y^{^{'}} + 64 y = 4 e^{- 4 x}

(D^{2} + 9) y = cos^{2} (x)

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

y^{^{''}} - y^{^{'}} - 6 y = 8 cos (3 x)

What is the solution for (d^2y)/(dx^2)+(dy)/(dx)-y=xsinh(x) ?
The solution for (d^2y)/(dx^2)+(dy)/(dx)-y=xsinh(x) is