What is the solution for y^{''}+2y^'-15y=-5x^2+3x-3 ?

The solution for y^{''}+2y^'-15y=-5x^2+3x-3 is y=c_{1}e^{3x}+c_{2}e^{-5x}+(x^2)/3-x/9+31/135

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y^{''}+2y^'-15y=-5x^2+3x-3

y^{^{''}} + 2 y^{^{'}} - 15 y = - 5 x^{2} + 3 x - 3

y = c_{1} e^{3 x} + c_{2} e^{- 5 x} + \frac{x ^{2}}{3} - \frac{x}{9} + \frac{31}{135}

Solution steps

y^{''} (x) + 2 y^{'} (x) - 15 y = - 5 x^{2} + 3 x - 3

Solve linear ODE:

y = c_{1} e^{3 x} + c_{2} e^{- 5 x} + \frac{x ^{2}}{3} - \frac{x}{9} + \frac{31}{135}

y = c_{1} e^{3 x} + c_{2} e^{- 5 x} + \frac{x ^{2}}{3} - \frac{x}{9} + \frac{31}{135}

a y^{^{''}} + b y = c

y^{^{''}} + 2 y^{^{'}} + 37 y = 4 H (t)

5 y^{^{''}} + 4 y^{^{'}} + 8 y = 4 x - 1

4 y^{^{''}} + 20 y^{^{'}} + 169 y = e^{2 t}

(D^{3} + 8) y = x^{4} + 2 x + 1

What is the solution for y^{''}+2y^'-15y=-5x^2+3x-3 ?
The solution for y^{''}+2y^'-15y=-5x^2+3x-3 is y=c_{1}e^{3x}+c_{2}e^{-5x}+(x^2)/3-x/9+31/135