What is the solution for x^{''}-10x^'+25x=78t^2e^{5t} ?

The solution for x^{''}-10x^'+25x=78t^2e^{5t} is x=c_{1}e^{5t}+c_{2}te^{5t}+(13e^{5t}t^4)/2

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x^{''}-10x^'+25x=78t^2e^{5t}

x^{^{''}} - 10 x^{^{'}} + 25 x = 78 t^{2} e^{5 t}

x = c_{1} e^{5 t} + c_{2} t e^{5 t} + \frac{13 e ^{5 t} t ^{4}}{2}

Solution steps

x^{''} (t) - 10 x^{'} (t) + 25 x = 78 t^{2} e^{5 t}

Solve linear ODE:

x = c_{1} e^{5 t} + c_{2} t e^{5 t} + \frac{13 e ^{5 t} t ^{4}}{2}

x = c_{1} e^{5 t} + c_{2} t e^{5 t} + \frac{13 e ^{5 t} t ^{4}}{2}

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

y^{^{''}} + 4 y^{^{'}} + 4 y = 6 e^{-} 2 x - 8 sin (x) + 6 cos (x)

y^{^{''}} - 4 y = e^{3 x}

y^{^{''}} - 9 y^{^{'}} + 20 y = 3 x + e^{4 x}

y^{^{''}} - 4 y^{^{'}} + 3 = e^{- 5}, y (0) = 0, y^{^{'}} (0) = 0

What is the solution for x^{''}-10x^'+25x=78t^2e^{5t} ?
The solution for x^{''}-10x^'+25x=78t^2e^{5t} is x=c_{1}e^{5t}+c_{2}te^{5t}+(13e^{5t}t^4)/2