What is the solution for y^{''}+y=e^x+x ?

The solution for y^{''}+y=e^x+x is y=c_{1}cos(x)+c_{2}sin(x)+1/2 e^x+x

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - y = x e^{x}

y^{^{'''}} - y^{^{''}} - y^{^{'}} + y = 2 e^{- t} + 3

y^{^{''}} + 2 y^{^{'}} + 10 y = x^{2} e^{- x} cos (3 x)

y^{^{''}} + 4 y = 9 t, y (0) = 0, y^{^{'}} (0) = 7

5 y^{^{''}} - 15 y = 10 e^{x}

What is the solution for y^{''}+y=e^x+x ?
The solution for y^{''}+y=e^x+x is y=c_{1}cos(x)+c_{2}sin(x)+1/2 e^x+x