What is the solution for y^{''}+y=3sin(2x)+5xcos(2x) ?

The solution for y^{''}+y=3sin(2x)+5xcos(2x) is y=c_{1}cos(x)+c_{2}sin(x)-(5xcos(2x))/3+11/9 sin(2x)

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y^{''}+y=3sin(2x)+5xcos(2x)

y^{^{''}} + y = 3 sin (2 x) + 5 x cos (2 x)

y = c_{1} cos (x) + c_{2} sin (x) - \frac{5 x cos ( 2 x )}{3} + \frac{11}{9} sin (2 x)

Solution steps

y^{''} (x) + y = 3 sin (2 x) + 5 x cos (2 x)

Solve linear ODE:

y = c_{1} cos (x) + c_{2} sin (x) - \frac{5 x cos ( 2 x )}{3} + \frac{11}{9} sin (2 x)

y = c_{1} cos (x) + c_{2} sin (x) - \frac{5 x cos ( 2 x )}{3} + \frac{11}{9} sin (2 x)

- 6 y^{^{''}} + 5 y^{^{'}} - 1 y = 1 t^{2} - 1 t + 6 e^{2} t

y^{^{''}} + 4 y^{^{'}} + 29 y = 200 sin (x)

\frac{d ^{2} y}{d x ^{2}} + 2 y = 6 x, y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} (p) + 6 y^{^{'}} (p) + 9 y (p) = p

y^{^{''}} - 2 y^{^{'}} - 35 y = 768 e^{- t}

What is the solution for y^{''}+y=3sin(2x)+5xcos(2x) ?
The solution for y^{''}+y=3sin(2x)+5xcos(2x) is y=c_{1}cos(x)+c_{2}sin(x)-(5xcos(2x))/3+11/9 sin(2x)