What is the solution for y^{''}-y=ln(t) ?

The solution for y^{''}-y=ln(t) is y=c_{1}e^t+c_{2}e^{-t}+(-2ln(t)+e^t\Ei(-t)+e^{-t}\Ei(t))/2

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y^{''}-y=ln(t)

y^{^{''}} - y = ln (t)

y = c_{1} e^{t} + c_{2} e^{- t} + \frac{- 2 ln ( t ) + e ^{t} Ei ( - t ) + e ^{- t} Ei ( t )}{2}

Solution steps

y^{''} (t) - y = ln (t)

Solve linear ODE:

y = c_{1} e^{t} + c_{2} e^{- t} + \frac{- 2 ln ( t ) + e ^{t} Ei ( - t ) + e ^{- t} Ei ( t )}{2}

y = c_{1} e^{t} + c_{2} e^{- t} + \frac{- 2 ln ( t ) + e ^{t} Ei ( - t ) + e ^{- t} Ei ( t )}{2}

y^{^{''}} - 12 y^{^{'}} + 36 y = 12 x + 5

y^{^{''}} + y = sin (x) + cos (x)

y^{^{''}} + 4 y = t^{2} + 2 e^{t}, y (0) = 0, y^{^{'}} (0) = 3

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

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

What is the solution for y^{''}-y=ln(t) ?
The solution for y^{''}-y=ln(t) is y=c_{1}e^t+c_{2}e^{-t}+(-2ln(t)+e^t\Ei(-t)+e^{-t}\Ei(t))/2