What is the solution for y^{''}-y=-6sin(x),y(0)=3,y^'(0)=2 ?

The solution for y^{''}-y=-6sin(x),y(0)=3,y^'(0)=2 is y=e^x+2e^{-x}+3sin(x)

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y^{''}-y=-6sin(x),y(0)=3,y^'(0)=2

y^{^{''}} - y = - 6 sin (x), y (0) = 3, y^{^{'}} (0) = 2

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - 2 y^{^{'}} + 5 y = e^{x} \cdot cos (2 x)

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

y^{^{''}} + 3 y^{^{'}} + 2 y = 13 + e x

y^{^{''}} + 10 y^{^{'}} + 25 y = 30 x + 3

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

What is the solution for y^{''}-y=-6sin(x),y(0)=3,y^'(0)=2 ?
The solution for y^{''}-y=-6sin(x),y(0)=3,y^'(0)=2 is y=e^x+2e^{-x}+3sin(x)