What is the solution for y^{''}+5y^'+4y=1+x+x^2 ?

The solution for y^{''}+5y^'+4y=1+x+x^2 is y=c_{1}e^{-x}+c_{2}e^{-4x}+(x^2)/4-(3x)/8+19/32

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y^{''}+5y^'+4y=1+x+x^2

y^{^{''}} + 5 y^{^{'}} + 4 y = 1 + x + x^{2}

y = c_{1} e^{- x} + c_{2} e^{- 4 x} + \frac{x ^{2}}{4} - \frac{3 x}{8} + \frac{19}{32}

Solution steps

y^{''} (x) + 5 y^{'} (x) + 4 y = 1 + x + x^{2}

Solve linear ODE:

y = c_{1} e^{- x} + c_{2} e^{- 4 x} + \frac{x ^{2}}{4} - \frac{3 x}{8} + \frac{19}{32}

y = c_{1} e^{- x} + c_{2} e^{- 4 x} + \frac{x ^{2}}{4} - \frac{3 x}{8} + \frac{19}{32}

y^{^{''}} + y = - x^{2}, y (0) = 0, y (1) = 1

y^{^{''}} + y^{^{'}} - 2 y = u π (t) - u^{2} π (t)

y^{^{''}} + 4 y^{^{'}} + 3 y = - 19 t e^{3 t}

y^{^{''}} + 4 y^{^{'}} = sin (t)

y^{^{''}} - 5 y^{^{'}} + 6 y = - 5 e^{3 x}

What is the solution for y^{''}+5y^'+4y=1+x+x^2 ?
The solution for y^{''}+5y^'+4y=1+x+x^2 is y=c_{1}e^{-x}+c_{2}e^{-4x}+(x^2)/4-(3x)/8+19/32