What is the solution for y^{''}-2y+y=3sin(2x)+2e^x ?

The solution for y^{''}-2y+y=3sin(2x)+2e^x is y=c_{1}e^x+c_{2}e^{-x}-3/5 sin(2x)+e^xx

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

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

y = c_{1} e^{x} + c_{2} e^{- x} - \frac{3}{5} sin (2 x) + e^{x} x

Solution steps

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

Solve linear ODE:

y = c_{1} e^{x} + c_{2} e^{- x} - \frac{3}{5} sin (2 x) + e^{x} x

y = c_{1} e^{x} + c_{2} e^{- x} - \frac{3}{5} sin (2 x) + e^{x} x

y^{^{'''}} - 7 y^{^{'}} - 6 y = sin (2 x) - 5

2 y^{^{''}} - y^{^{'}} - 6 y = - 7 e^{2 x}, y (0) = 5, y^{^{'}} (0) = 2

y^{^{''}} + y^{^{'}} + y = 1, y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} + 3 y^{^{'}} - 10 = 0, y (0) = 0, y^{^{'}} (0) = 0

\frac{d ^{2} y}{d x ^{2}} + \frac{d}{d x} (8 y) + 65 y = 100

What is the solution for y^{''}-2y+y=3sin(2x)+2e^x ?
The solution for y^{''}-2y+y=3sin(2x)+2e^x is y=c_{1}e^x+c_{2}e^{-x}-3/5 sin(2x)+e^xx