What is the solution for y^{''}-9y^'+20y=e^x(-9-78x+36x^2) ?

The solution for y^{''}-9y^'+20y=e^x(-9-78x+36x^2) is y=c_{1}e^{5x}+c_{2}e^{4x}+3x^2e^x-3e^xx-3e^x

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y^{''}-9y^'+20y=e^x(-9-78x+36x^2)

y^{^{''}} - 9 y^{^{'}} + 20 y = e^{x} (- 9 - 78 x + 36 x^{2})

y = c_{1} e^{5 x} + c_{2} e^{4 x} + 3 x^{2} e^{x} - 3 e^{x} x - 3 e^{x}

Solution steps

y^{''} (x) - 9 y^{'} (x) + 20 y = e^{x} (- 9 - 78 x + 36 x^{2})

Solve linear ODE:

y = c_{1} e^{5 x} + c_{2} e^{4 x} + 3 x^{2} e^{x} - 3 e^{x} x - 3 e^{x}

y = c_{1} e^{5 x} + c_{2} e^{4 x} + 3 x^{2} e^{x} - 3 e^{x} x - 3 e^{x}

y^{^{''}} - 3 y^{^{'}} - 10 y = - 18 t + 20 t^{2}

y^{^{'''}} - y^{^{'}} = e^{t} (1 + t)

(d^{2} - 2 d + 5) y = e^{2 x} \cdot sin (x)

\frac{d ^{2} y}{d x ^{2}} - 5 \frac{d y}{d x} + 6 y = x^{3} + 2 x^{2} + 5 x + 2

y^{^{''}} + 9 y = 2 x^{2} e^{3 x} + 5

What is the solution for y^{''}-9y^'+20y=e^x(-9-78x+36x^2) ?
The solution for y^{''}-9y^'+20y=e^x(-9-78x+36x^2) is y=c_{1}e^{5x}+c_{2}e^{4x}+3x^2e^x-3e^xx-3e^x