How do you solve a non-linear ODE involving a variable mass dust particle?

[Matt_993]

Homework Statement

At time t=0 a dust particle of mass m_0 starts to fall from rest through a cloud. Its mass grows exponentially with the distance fallen, so that after falling through a distance x its mass is m_0exp[αx] where α is constant. Show that at time t the velocity of the particle is given by:

v=sqrt(g/α)tanh(t(sqrt(αg))

Homework Equations

Variable mass equation:

mg= mv'+vm'

The Attempt at a Solution

Using the variable mass equation I've fiddled around with it and gotten out a differential equation in terms of v, which is:

dv/dt=-αv^2+g

Which after filling in the answer I am looking for is the correct equation. However solving it properly by hand seems harder as its a non-linear ODE. I was just wondering is there any other way of going about this problem or if not, can anyone give me a tip on how to solve the ODE properly.

Thank you for your time

 

[TSny]

Hello, Matt_993. Welcome to PF!

Using the variable mass equation I've fiddled around with it and gotten out a differential equation in terms of v, which is:

dv/dt=-αy^2+g

Did you mistype this? What does the symbol y stand for?

Once you get the equation written in terms of just 2 variables, you might try the technique of separation of variables.

 

[Matt_993]

Yea the y should've been a v, I've changed it. But I can't seem to be able to separate the variables in the equation because of g, is there another way?

 

[TSny]

Yea the y should've been a v, I've changed it. But I can't seem to be able to separate the variables in the equation because of g, is there another way?

Divide both sides of the equation by the entire right hand side.

 

[Matt_993]

How very stupid of me, can't believe I missed that.

That works, thanks for the help :)