A particular solution of $$\ddot x -3\dot x 2x = 2 e^{-2t}$$ is given by

A particular solution of $$\ddot x -\dot x - 2x = 2 \sin{2t}$$ is given by

A particular solution of $$\ddot x -3\dot x 2x = t^2 1$$ is given by

The solution of $$\ddot x 2\dot x 2x = \cos{t}$$ with initial conditions $$x(0)=0$$ and $$\dot x(0)=0$$ is given by

The solution of $$\ddot x = kt$$ with initial conditions $$x(0)=1$$ and $$\dot x(0)=-1$$ is given by

The solution of $$\ddot x \omega^2 x = f\sin{\omega t}$$ with initial conditions $$x(0)=0$$ and $$\dot x(0)=0$$ is given by

The solution of the differential equation given by

$$\ddot x \dot x = e^{-t},$$

with $$x(0)=0$$ and $$\dot x(0)=0$$, is given by

The solution of the differential equation given by

$$\ddot x - x = \sinh{t}$$,

with $$x(0)=0$$ and $$\dot x(0)=0$$, is given by

When comparing the $$RLC$$ circuit to the mass on a spring and to the pendulum, the inductor $$L$$ plays the role of a

The $$RLC$$ circuit equation, given by

$$\displaystyle L\frac{d^2q}{dt^2} R\frac{dq}{dt} \frac{1}{C}q = \mathcal{E}_0 \cos{\omega t},$$

can be put in the dimensionless form

$$\displaystyle \alpha \frac{d^2Q}{d\tau^2} \frac{dQ}{d\tau} Q = \cos{\beta \tau}$$.

The dimensionless product $$\alpha \beta$$ is equal to