What is the solution for (d^2y)/(dt^2)-6(dy)/(dt)+9y=t ?

The solution for (d^2y)/(dt^2)-6(dy)/(dt)+9y=t is y=c_{1}e^{3t}+c_{2}te^{3t}+t/9+2/27

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(d^2y)/(dt^2)-6(dy)/(dt)+9y=t

\frac{d ^{2} y}{d t ^{2}} - 6 \frac{d y}{d t} + 9 y = t

y = c_{1} e^{3 t} + c_{2} t e^{3 t} + \frac{t}{9} + \frac{2}{27}

Solution steps

\frac{d ^{2} y}{d t ^{2}} - 6 \frac{d y}{d t} + 9 y = t

Solve linear ODE:

y = c_{1} e^{3 t} + c_{2} t e^{3 t} + \frac{t}{9} + \frac{2}{27}

y = c_{1} e^{3 t} + c_{2} t e^{3 t} + \frac{t}{9} + \frac{2}{27}

y^{^{'''}} + 9 y^{^{'}} = 27 x^{2}

4 y^{^{''}} - y = e^{\frac{x}{2}}

y^{^{'''}} + 9 y^{^{''}} + 27 y^{^{'}} + 27 y = 5 e^{- 3 x}

(D^{2} - 5 D + 6) y = 3 e^{3 x} + e^{2 x}

y^{^{''}} - y^{^{'}} - 2 y = 3 cos (2 x)

What is the solution for (d^2y)/(dt^2)-6(dy)/(dt)+9y=t ?
The solution for (d^2y)/(dt^2)-6(dy)/(dt)+9y=t is y=c_{1}e^{3t}+c_{2}te^{3t}+t/9+2/27