What is the solution for y^{''}+y=5e^{-4x},y(0)=0,y^'(0)=0 ?

The solution for y^{''}+y=5e^{-4x},y(0)=0,y^'(0)=0 is y=-5/17 cos(x)+20/17 sin(x)+5/17 e^{-4x}

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

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

y = - \frac{5}{17} cos (x) + \frac{20}{17} sin (x) + \frac{5}{17} e^{- 4 x}

Solution steps

y^{''} (x) + y = 5 e^{- 4 x}

Solve linear ODE:

y = - \frac{5}{17} cos (x) + \frac{20}{17} sin (x) + \frac{5}{17} e^{- 4 x}

y = - \frac{5}{17} cos (x) + \frac{20}{17} sin (x) + \frac{5}{17} e^{- 4 x}

6 y^{^{''}} - y^{^{'}} - 2 y = 3 sin (10 x)

y^{^{''}} - 7 y^{^{'}} + 12 y = 5 x - 8

y^{^{''}} + y = 10 sin (2 t) + 7 t cos (2 t)

y^{^{''}} - 10 y^{^{'}} + 25 y = 6 e^{5 x}

y^{^{''}} + 8 y^{^{'}} + 16 y = - 5 x e^{- 3 x} + 3 e^{- 4 x} - 4

What is the solution for y^{''}+y=5e^{-4x},y(0)=0,y^'(0)=0 ?
The solution for y^{''}+y=5e^{-4x},y(0)=0,y^'(0)=0 is y=-5/17 cos(x)+20/17 sin(x)+5/17 e^{-4x}