What is the solution for (d^2y)/(dx^2)+(dy)/(dx)=x ?

The solution for (d^2y)/(dx^2)+(dy)/(dx)=x is y=c_{1}+c_{2}e^{-x}+(x^2)/2-x

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(d^2y)/(dx^2)+(dy)/(dx)=x

\frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} = x

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

Solution steps

\frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} = x

Solve linear ODE:

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

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

y^{^{''}} + 4 y + 4 y = (3 + x) e^{- 2 x}

y^{^{''}} - 4 y^{^{'}} + 3 y = 3 x - 1

y^{^{''}} - 6 y^{^{'}} + 8 y = 3 e^{- 2 x} + 2 x

y^{^{''}} + 4 y = 8 cos (2 x) + 12 sin (2 x)

x^{^{''}} + 4 x^{^{'}} + 3 x = 1

What is the solution for (d^2y)/(dx^2)+(dy)/(dx)=x ?
The solution for (d^2y)/(dx^2)+(dy)/(dx)=x is y=c_{1}+c_{2}e^{-x}+(x^2)/2-x