What is the solution for y^{''}+y=3e^{-6x},y(0)=0,y^'(0)=0 ?

The solution for y^{''}+y=3e^{-6x},y(0)=0,y^'(0)=0 is y=-3/37 cos(x)+18/37 sin(x)+3/37 e^{-6x}

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y^{''}+y=3e^{-6x},y(0)=0,y^'(0)=0

y^{^{''}} + y = 3 e^{- 6 x}, y (0) = 0, y^{^{'}} (0) = 0

y = - \frac{3}{37} cos (x) + \frac{18}{37} sin (x) + \frac{3}{37} e^{- 6 x}

Solution steps

y^{''} (x) + y = 3 e^{- 6 x}

Solve linear ODE:

y = - \frac{3}{37} cos (x) + \frac{18}{37} sin (x) + \frac{3}{37} e^{- 6 x}

y = - \frac{3}{37} cos (x) + \frac{18}{37} sin (x) + \frac{3}{37} e^{- 6 x}

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

y^{^{''}} - 6 y^{^{'}} + 9 y = t^{2} e^{3} t

y^{^{''}} - 10 y^{^{'}} + 25 y = 132 x^{2} (e^{(5 x)})

y^{^{''}} - 2 y = e^{2 x} + x^{2} - 1

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

What is the solution for y^{''}+y=3e^{-6x},y(0)=0,y^'(0)=0 ?
The solution for y^{''}+y=3e^{-6x},y(0)=0,y^'(0)=0 is y=-3/37 cos(x)+18/37 sin(x)+3/37 e^{-6x}