What is the solution for (d^2y)/(dx^2)+ay^'=b ?

The solution for (d^2y)/(dx^2)+ay^'=b is y=c_{1}+c_{2}e^{-ax}+(bx)/a

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(d^2y)/(dx^2)+ay^'=b

\frac{d ^{2} y}{d x ^{2}} + a y^{^{'}} = b

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

Solution steps

y^{''} (x) + a y^{'} (x) = b

Solve linear ODE:

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

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

y^{^{''}} + y = 2 sin (2) t

y^{^{''}} - 2 y^{^{'}} + y = e^{x} + x

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

y^{^{''}} - 81 y = cosh (9 x)

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

What is the solution for (d^2y)/(dx^2)+ay^'=b ?
The solution for (d^2y)/(dx^2)+ay^'=b is y=c_{1}+c_{2}e^{-ax}+(bx)/a