What is the solution for 2y^{''}-5y^'-3y=3-x ?

The solution for 2y^{''}-5y^'-3y=3-x is y=c_{1}e^{3x}+c_{2}e^{-x/2}+x/3-14/9

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2y^{''}-5y^'-3y=3-x

2 y^{^{''}} - 5 y^{^{'}} - 3 y = 3 - x

y = c_{1} e^{3 x} + c_{2} e^{- \frac{x}{2}} + \frac{x}{3} - \frac{14}{9}

Solution steps

2 y^{''} (x) - 5 y^{'} (x) - 3 y = 3 - x

Solve linear ODE:

y = c_{1} e^{3 x} + c_{2} e^{- \frac{x}{2}} + \frac{x}{3} - \frac{14}{9}

y = c_{1} e^{3 x} + c_{2} e^{- \frac{x}{2}} + \frac{x}{3} - \frac{14}{9}

y^{^{''}} - 3 y^{^{'}} + 2 y = 4 t + e^{3 t}

2 y^{^{''}} - 8 y^{^{'}} - 4 y = - 5 x^{2} - 12 x e^{- x}

y^{^{''}} - 5 y^{^{'}} - 24 y = (t + 1) e^{t}

5 y^{^{''}} + y^{^{'}} = - 4 x, y (0) = 0, y^{^{'}} (0) = - 25

\frac{d ^{2} x}{d t ^{2}} + \frac{d x}{d t} = 5 t

What is the solution for 2y^{''}-5y^'-3y=3-x ?
The solution for 2y^{''}-5y^'-3y=3-x is y=c_{1}e^{3x}+c_{2}e^{-x/2}+x/3-14/9