What is the solution for y^{''}+4y=e^t,y(0)=0,y^'(0)=0 ?

The solution for y^{''}+4y=e^t,y(0)=0,y^'(0)=0 is y=-1/5 cos(2t)-1/10 sin(2t)+1/5 e^t

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y^{''}+4y=e^t,y(0)=0,y^'(0)=0

y^{^{''}} + 4 y = e^{t}, y (0) = 0, y^{^{'}} (0) = 0

y = - \frac{1}{5} cos (2 t) - \frac{1}{10} sin (2 t) + \frac{1}{5} e^{t}

Solution steps

y^{''} (t) + 4 y = e^{t}

Solve linear ODE:

y = - \frac{1}{5} cos (2 t) - \frac{1}{10} sin (2 t) + \frac{1}{5} e^{t}

y = - \frac{1}{5} cos (2 t) - \frac{1}{10} sin (2 t) + \frac{1}{5} e^{t}

What is the solution for y^{''}+4y=e^t,y(0)=0,y^'(0)=0 ?
The solution for y^{''}+4y=e^t,y(0)=0,y^'(0)=0 is y=-1/5 cos(2t)-1/10 sin(2t)+1/5 e^t