Second order differential equation with constant second member $-g$

Question

I would like to solve this differential equation :
$$\ddot z(t)+A\dot z(t)=-g$$
where $g$ is constant.
I know that the solution of :
$$\ddot z(t)+A\dot z(t)=0$$
is $z(t)=\alpha + \beta e^{-At}$ but I can't find the particular solution…

Answer 1

Let $u(t)=\dot z(t)$ then your ODE is of first order with respect to $u$: $$\dot u(t)+A u(t)=-g$$ where the solution is $u(t)=\beta e^{-At}-\frac{g}{A}$ (the particular solution is easy to find here).

Finally, in order to find $z$, we integrate $u$ with respect to $t$: $$z(t)=\int u(t)\,dt=\int\left(\beta e^{-At}-\frac{g}{A}\right)dt=-\beta e^{-At}-\frac{gt}{A}+\alpha.$$ where $\alpha$ and $\beta$ are arbitrary real constants.