What is the solution for y^{''}+y=8sin(x)+4cos(x)+1 ?

The solution for y^{''}+y=8sin(x)+4cos(x)+1 is y=c_{1}cos(x)+c_{2}sin(x)-4xcos(x)+2xsin(x)+1

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y^{''}+y=8sin(x)+4cos(x)+1

y^{^{''}} + y = 8 sin (x) + 4 cos (x) + 1

y = c_{1} cos (x) + c_{2} sin (x) - 4 x cos (x) + 2 x sin (x) + 1

Solution steps

y^{''} (x) + y = 8 sin (x) + 4 cos (x) + 1

Solve linear ODE:

y = c_{1} cos (x) + c_{2} sin (x) - 4 x cos (x) + 2 x sin (x) + 1

y = c_{1} cos (x) + c_{2} sin (x) - 4 x cos (x) + 2 x sin (x) + 1

x^{^{''}} + 4 x^{^{'}} + 4 = 0

y^{^{''}} - 2 y^{^{'}} + y = \frac{e ^{t}}{( 1 + t ^{2} ) + 3 e ^{t}}

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

y^{^{''}} + 2 y^{^{'}} + 2 y = 4 cos (t) + 2 sin (t)

y^{^{''}} + 10 y^{^{'}} + 25 y = 100 sinh (5 x)

What is the solution for y^{''}+y=8sin(x)+4cos(x)+1 ?
The solution for y^{''}+y=8sin(x)+4cos(x)+1 is y=c_{1}cos(x)+c_{2}sin(x)-4xcos(x)+2xsin(x)+1