What is the solution for y^{''}+81y=2sin(2x),y(0)=1,y^'(0)=0 ?

The solution for y^{''}+81y=2sin(2x),y(0)=1,y^'(0)=0 is y=cos(9x)-4/693 sin(9x)+2/77 sin(2x)

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y^{''}+81y=2sin(2x),y(0)=1,y^'(0)=0

y^{^{''}} + 81 y = 2 sin (2 x), y (0) = 1, y^{^{'}} (0) = 0

y = cos (9 x) - \frac{4}{693} sin (9 x) + \frac{2}{77} sin (2 x)

Solution steps

y^{''} (x) + 81 y = 2 sin (2 x)

Solve linear ODE:

y = cos (9 x) - \frac{4}{693} sin (9 x) + \frac{2}{77} sin (2 x)

y = cos (9 x) - \frac{4}{693} sin (9 x) + \frac{2}{77} sin (2 x)

y^{^{''}} - y = x, y (0) = 0, y (1) = 0

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

y^{^{''}} - 2 y = sin (t)

y^{^{''}} + 2 y^{^{'}} + 5 y = 34 cos (2 x)

(D^{2} - 7 D + 6) y = e^{6 x} + e^{x}

What is the solution for y^{''}+81y=2sin(2x),y(0)=1,y^'(0)=0 ?
The solution for y^{''}+81y=2sin(2x),y(0)=1,y^'(0)=0 is y=cos(9x)-4/693 sin(9x)+2/77 sin(2x)