Simple problem in Mechanics, weird differential equation

[dujardin]

There is a problem I couldn't figure out , it says :

it says that a particle of mass m moves along a straight line and is acted on by a retarding force (one always directed against the motion) F=b*exp(a*v(t)),
b, a are constants and v is the velocity.

At t=0 it is moving with velocity V

and I am aked to solve the differential equation that results from this to get a function of v(t).

I found that the differential equation that has to be solved is :

dv/dt = (b/m)*exp[a*v]

so this is like solving a non-linear differential equation of the form y'=exp(y)

How do you do that??

 

[dextercioby]

Do you know how to separate variables ?

 

[jambaugh]

You have simply to re-write the equation in terms of differentials and you will be able to integrate:

$$\frac{du}{dt} = f(u)$$
becomes:
$$du = f(u) dt$$
and then
$$\frac{du}{f(u)} = dt$$
you can then integrate:
$$\int \frac{du}{f(u)} = t + C$$
which gives you $$u(t)$$ implicitly.

In your case you should get $$v(t)$$ in terms of a logarithm of t.

 

[dujardin]

You have simply to re-write the equation in terms of differentials and you will be able to integrate:

$$\frac{du}{dt} = f(u)$$
becomes:
$$du = f(u) dt$$
and then
$$\frac{du}{f(u)} = dt$$
you can then integrate:
$$\int \frac{du}{f(u)} = t + C$$
which gives you $$u(t)$$ implicitly.

In your case you should get $$v(t)$$ in terms of a logarithm of t.

I understood everything. Thank you so much

 

[Rainbow Child]

Just a note!

Since you have a retarding force, the ODE should be

$$\frac{d\,v}{d\,t}=-\frac{b}{m}\,e^{a\,v}$$