What is the solution for y^{''}-8y^'+16y=45e^{7t} ?

The solution for y^{''}-8y^'+16y=45e^{7t} is y=c_{1}e^{4t}+c_{2}te^{4t}+5e^{7t}

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y^{''}-8y^'+16y=45e^{7t}

y^{^{''}} - 8 y^{^{'}} + 16 y = 45 e^{7 t}

y = c_{1} e^{4 t} + c_{2} t e^{4 t} + 5 e^{7 t}

Solution steps

y^{''} (t) - 8 y^{'} (t) + 16 y = 45 e^{7 t}

Solve linear ODE:

y = c_{1} e^{4 t} + c_{2} t e^{4 t} + 5 e^{7 t}

y = c_{1} e^{4 t} + c_{2} t e^{4 t} + 5 e^{7 t}

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

y^{^{''}} - 8 y^{^{'}} + 16 y = 2 e^{4 x}

y^{^{''}} + 4 y^{^{'}} + 4 y = e^{x} cos (x)

y^{^{''}} + 4 y^{^{'}} + 13 y = 5 cos (2 t)

x^{^{''}} - 6 x^{^{'}} + 10 x = 30 t

What is the solution for y^{''}-8y^'+16y=45e^{7t} ?
The solution for y^{''}-8y^'+16y=45e^{7t} is y=c_{1}e^{4t}+c_{2}te^{4t}+5e^{7t}