I have the following differential equation
$my'' + \beta y' + mg = 0$ , with condition $y(0)=0$.
I need to solve the equation and obtain expressions for the unknowns.
I have attempted to use the try method of substituting in $y=e^{\lambda t}$ but do not know how to proceed beyond that.
The differential equation is meant to represent free-fall of an object with air resistance that is proportional to it's speed.
Construct $q(t)=y(t)+\frac{mg}{\beta}t$
By substitution
$$ mq''+\beta q'=0 $$
The solutions of above ODE are
$$ q(t)=c$$ Or $$ q(t)=c_1e^{-\frac{\beta}{m}t}+c_2$$
$c,c_1,c_2$ are all constants
Therefore, with initial condition $y(0)=0$,the solutions of original ODE are
$$ y(t) =-\frac{mg}{\beta}t$$ Or $$y(t) = c(e^{-\frac{\beta}{m}t}-1)-\frac{mg}{\beta}t$$