What is the solution for x^{''}-2x^'+x=2000t^2-150e^t ?

The solution for x^{''}-2x^'+x=2000t^2-150e^t is x=c_{1}e^t+c_{2}te^t+2000t^2+8000t+12000-75t^2e^t

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x^{''}-2x^'+x=2000t^2-150e^t

x^{^{''}} - 2 x^{^{'}} + x = 2000 t^{2} - 150 e^{t}

x = c_{1} e^{t} + c_{2} t e^{t} + 2000 t^{2} + 8000 t + 12000 - 75 t^{2} e^{t}

Solution steps

x^{''} (t) - 2 x^{'} (t) + x = 2000 t^{2} - 150 e^{t}

Solve linear ODE:

x = c_{1} e^{t} + c_{2} t e^{t} + 2000 t^{2} + 8000 t + 12000 - 75 t^{2} e^{t}

x = c_{1} e^{t} + c_{2} t e^{t} + 2000 t^{2} + 8000 t + 12000 - 75 t^{2} e^{t}

y^{^{'''}} + 10 y^{^{''}} + 25 y^{^{'}} = e^{x}

y^{^{''}} + 4 y = (x^{2} - 3) sin (2 x)

y^{^{''}} - 5 y^{^{'}} - 6 y = 2 cos (3 x) + 4 e^{2 x}

(D^{2} - D - 2) y = e^{3 x}

y^{^{''}} - 3 y^{^{'}} + 2 y = x e^{2 x}

What is the solution for x^{''}-2x^'+x=2000t^2-150e^t ?
The solution for x^{''}-2x^'+x=2000t^2-150e^t is x=c_{1}e^t+c_{2}te^t+2000t^2+8000t+12000-75t^2e^t