What is the solution for y^{''}+y=6e^{-7x},y(0)=0,y^'(0)=0 ?

The solution for y^{''}+y=6e^{-7x},y(0)=0,y^'(0)=0 is y=-3/25 cos(x)+21/25 sin(x)+3/25 e^{-7x}

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

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

y = - \frac{3}{25} cos (x) + \frac{21}{25} sin (x) + \frac{3}{25} e^{- 7 x}

Solution steps

y^{''} (x) + y = 6 e^{- 7 x}

Solve linear ODE:

y = - \frac{3}{25} cos (x) + \frac{21}{25} sin (x) + \frac{3}{25} e^{- 7 x}

y = - \frac{3}{25} cos (x) + \frac{21}{25} sin (x) + \frac{3}{25} e^{- 7 x}

y^{^{''}} + 4 y^{^{'}} - 5 y = t e^{t}, y (0) = 1, y^{^{'}} (0) = 0

y^{^{''}} - 4 y^{^{'}} - 12 y = x - 6

x^{^{''}} + x = cos (3 t)

y^{^{'''}} - 5 y^{^{''}} + 6 y^{^{'}} = 5 + 4 sin (x)

y^{^{''}} + 2 y^{^{'}} - 4 y = 8 x^{2}

What is the solution for y^{''}+y=6e^{-7x},y(0)=0,y^'(0)=0 ?
The solution for y^{''}+y=6e^{-7x},y(0)=0,y^'(0)=0 is y=-3/25 cos(x)+21/25 sin(x)+3/25 e^{-7x}