What is the solution for (d^2x)/(dt^2)+2(dx)/(dt)+x=3cos(t) ?

The solution for (d^2x)/(dt^2)+2(dx)/(dt)+x=3cos(t) is x=c_{1}e^{-t}+c_{2}te^{-t}+3/2 sin(t)

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(d^2x)/(dt^2)+2(dx)/(dt)+x=3cos(t)

\frac{d ^{2} x}{d t ^{2}} + 2 \frac{d x}{d t} + x = 3 cos (t)

x = c_{1} e^{- t} + c_{2} t e^{- t} + \frac{3}{2} sin (t)

Solution steps

\frac{d ^{2} x}{d t ^{2}} + 2 \frac{d x}{d t} + x = 3 cos (t)

Solve linear ODE:

x = c_{1} e^{- t} + c_{2} t e^{- t} + \frac{3}{2} sin (t)

x = c_{1} e^{- t} + c_{2} t e^{- t} + \frac{3}{2} sin (t)

y^{^{''}} - 18 y = tan (x)

8 y^{^{''}} + 0.25 y^{^{'}} + 512 y = 2 cos (2 t)

y^{^{''}} + y = 1, 6 cos (t) y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} - y = 15 t, y (0) = - 15 t

x^{^{''}} + 6 x^{^{'}} + 5 x = sin (t)

What is the solution for (d^2x)/(dt^2)+2(dx)/(dt)+x=3cos(t) ?
The solution for (d^2x)/(dt^2)+2(dx)/(dt)+x=3cos(t) is x=c_{1}e^{-t}+c_{2}te^{-t}+3/2 sin(t)