What is the solution for y^{''}+10y^'+20y=50sinh(5x) ?

The solution for y^{''}+10y^'+20y=50sinh(5x) is

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y^{''}+10y^'+20y=50sinh(5x)

y^{^{''}} + 10 y^{^{'}} + 20 y = 50 sinh (5 x)

y = c_{1} e^{(- 5 + 5) x} + c_{2} e^{(- 5 - 5) x} + e^{- 5 x + 5 x} \cdot \int 55 e^{5 x - 5 x} sinh (5 x) d x + e^{- 5 x - 5 x} \cdot \int - 55 e^{5 x + 5 x} sinh (5 x) d x

Solution steps

y^{''} (x) + 10 y^{'} (x) + 20 y = 50 sinh (5 x)

Solve linear ODE:

y = c_{1} e^{(- 5 + 5) x} + c_{2} e^{(- 5 - 5) x} + e^{- 5 x + 5 x} \cdot \int 55 e^{5 x - 5 x} sinh (5 x) d x + e^{- 5 x - 5 x} \cdot \int - 55 e^{5 x + 5 x} sinh (5 x) d x

y = c_{1} e^{(- 5 + 5) x} + c_{2} e^{(- 5 - 5) x} + e^{- 5 x + 5 x} \cdot \int 55 e^{5 x - 5 x} sinh (5 x) d x + e^{- 5 x - 5 x} \cdot \int - 55 e^{5 x + 5 x} sinh (5 x) d x

\frac{d ^{4} y}{d x ^{4}} + 3 \frac{d ^{2} y}{d x ^{2}} - 4 = 0

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

y^{^{''}} + 17 y = 5 e^{x}

y^{^{''}} + 3 y^{^{'}} + 2 y = \frac{1}{( 9 + e ^{x} )}

y^{^{''}} + 10 y^{^{'}} + 25 y = 128 e^{3 t}

What is the solution for y^{''}+10y^'+20y=50sinh(5x) ?
The solution for y^{''}+10y^'+20y=50sinh(5x) is