How Do You Isolate Variables in a Complex Differential Equation?

[stunner5000pt]

can you help me olve this (simple?) differential equation
$$m \ddot{y} = -mg - \beta \dot{y}$$
integrate once and i get
$$\dot{y} = -gt - \frac{\beta y}{m} + C$$
also can be written as
$$\frac{dy}{dt} = -gt - \frac{\beta y}{m} + C$$
both are equivalent

basically trying to get the y on one side adn the t on the other side. HJva tried many ways but can't isolate the two. Any suggestions?

 

[berkeman]

Have you tried the approach where you assume a solution for y(t), differentiate it twice and work out the equation constants...? What would be a typical function that could work with this approach...?

 

[vladb]

First of all let v = dy/dt, then you have
$m \frac{dv}{dt} = -mg - \beta v$
From which

$\frac{dv}{g + \frac{\beta}{m} v} = -dt$

Integrating it now gives

$\frac{m}{\beta} \ln(g + \frac{\beta}{m} v) = -t + C_1$

Solving for v gives:

$v = C e^{-\frac{\beta}{m}t} - \frac{mg}{\beta}$

Where C is some new constant which is I think $C = \frac{m}{\beta}e^{\frac{\beta}{m} C_1}$. But it doesn't matter though.

Now remember that v = dy/dt. Thus integrating the last equation will give you y(t). (there will be two constants then)