What is the solution for (d^2y)/(dt^2)=1-e^{2t} ?

The solution for (d^2y)/(dt^2)=1-e^{2t} is y=(t^2)/2-1/4 e^{2t}+c_{1}t+c_{2}

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(d^2y)/(dt^2)=1-e^{2t}

\frac{d ^{2} y}{d t ^{2}} = 1 - e^{2 t}

y = \frac{t ^{2}}{2} - \frac{1}{4} e^{2 t} + c_{1} t + c_{2}

Solution steps

\frac{d ^{2} y}{d t ^{2}} = 1 - e^{2 t}

Substitute

\frac{d ^{2} y}{d t ^{2}}

with

y^{''} (t)

y^{^{''}} (t) = 1 - e^{2 t}

Solve

y^{''} (t) = 1 - e^{2 t} : y^{'} (t) = t - \frac{1}{2} e^{2 t} + c_{1}

Solve

y^{'} (t) = t - \frac{1}{2} e^{2 t} + c_{1} : y = \frac{t ^{2}}{2} - \frac{1}{4} e^{2 t} + c_{1} t + c_{2}

y = \frac{t ^{2}}{2} - \frac{1}{4} e^{2 t} + c_{1} t + c_{2}

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What is the solution for (d^2y)/(dt^2)=1-e^{2t} ?
The solution for (d^2y)/(dt^2)=1-e^{2t} is y=(t^2)/2-1/4 e^{2t}+c_{1}t+c_{2}