What is the solution for (d^2y)/(dx^2)+y=e^x,y(0)=0,y^'(0)=0 ?

The solution for (d^2y)/(dx^2)+y=e^x,y(0)=0,y^'(0)=0 is y=-1/2 cos(x)-1/2 sin(x)+1/2 e^x

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(d^2y)/(dx^2)+y=e^x,y(0)=0,y^'(0)=0

\frac{d ^{2} y}{d x ^{2}} + y = e^{x}, y (0) = 0, y^{^{'}} (0) = 0

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} + y = t, y (0) = 1, y^{^{'}} (0) = - 2

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

y^{^{''}} - 2 y^{^{'}} - 48 y = - 136 t + 96 t^{2}

y^{^{''}} + 15 y^{^{'}} + 56 y = e^{- 6 x}

y^{^{''}} + 4 y = 6 cos (2 x)

What is the solution for (d^2y)/(dx^2)+y=e^x,y(0)=0,y^'(0)=0 ?
The solution for (d^2y)/(dx^2)+y=e^x,y(0)=0,y^'(0)=0 is y=-1/2 cos(x)-1/2 sin(x)+1/2 e^x