What is the solution for y^{''}+y=8e^{-6x},y(0)=0,y^'(0)=0 ?

The solution for y^{''}+y=8e^{-6x},y(0)=0,y^'(0)=0 is y=-8/37 cos(x)+48/37 sin(x)+8/37 e^{-6x}

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y^{''}+y=8e^{-6x},y(0)=0,y^'(0)=0

y^{^{''}} + y = 8 e^{- 6 x}, y (0) = 0, y^{^{'}} (0) = 0

y = - \frac{8}{37} cos (x) + \frac{48}{37} sin (x) + \frac{8}{37} e^{- 6 x}

Solution steps

y^{''} (x) + y = 8 e^{- 6 x}

Solve linear ODE:

y = - \frac{8}{37} cos (x) + \frac{48}{37} sin (x) + \frac{8}{37} e^{- 6 x}

y = - \frac{8}{37} cos (x) + \frac{48}{37} sin (x) + \frac{8}{37} e^{- 6 x}

y^{^{''}} - 6 y^{^{'}} - 6 y = 2 x e^{- x}

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

y^{^{''}} + y^{^{'}} - 1 = 0

y^{^{''}} - 4 y^{^{'}} + 13 y = 4 + 3 e^{t}

y^{^{''}} - y = 2 (e^{x} - e^{- x})^{- 1}

What is the solution for y^{''}+y=8e^{-6x},y(0)=0,y^'(0)=0 ?
The solution for y^{''}+y=8e^{-6x},y(0)=0,y^'(0)=0 is y=-8/37 cos(x)+48/37 sin(x)+8/37 e^{-6x}