What is the solution for y^{''}+y=3sin(2t)+tcos(2t) ?

The solution for y^{''}+y=3sin(2t)+tcos(2t) is y=c_{1}cos(t)+c_{2}sin(t)-(tcos(2t))/3-5/9 sin(2t)

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y^{''}+y=3sin(2t)+tcos(2t)

y^{^{''}} + y = 3 sin (2 t) + t cos (2 t)

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

Solution steps

y^{''} (t) + y = 3 sin (2 t) + t cos (2 t)

Solve linear ODE:

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

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

\frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} - 2 y = 2 x

y^{^{''}} - 2 y^{^{'}} + y = \frac{11 e ^{t}}{1 + t ^{2}}

(D^{2} + 4) y = sin (2 x)

y^{^{''}} - 5 y^{^{'}} + 6 y = 6 x + 2 - 3 e^{2 x}

y^{^{'''}} - 4 y^{^{''}} + 6 y^{^{'}} - 4 y = x + e^{x} cos (x)

What is the solution for y^{''}+y=3sin(2t)+tcos(2t) ?
The solution for y^{''}+y=3sin(2t)+tcos(2t) is y=c_{1}cos(t)+c_{2}sin(t)-(tcos(2t))/3-5/9 sin(2t)