Integrating ODE: dv/dt=g-kv/m Step-by-Step Guide

[Math10]

Homework Statement

How to integrate dv/dt=g-kv/m?

Homework Equations

This is ODE.

The Attempt at a Solution

dv/(g-(kv/m))=dt
Now what? I don't know how to integrate the left side.

 

[Mark44]

Homework Statement

How to integrate dv/dt=g-kv/m?

Homework Equations

This is ODE.

The Attempt at a Solution

dv/(g-(kv/m))=dt
Now what? I don't know how to integrate the left side.

The left side can be rewritten as $\frac{m \ dv}{mg - kv}$. There's a fairly obvious substitution you can use.

 

[Math10]

What substitution?

 

[Math10]

Never mind. I integrated and got (-m/k)ln abs(gm-kv)+C.

 

[Math10]

Now, how to simplify (-m/k)ln abs(gm-kv)=t+C?

 

[Stephen Tashi]

dv/(g-(kv/m))=dt
Now what? I don't know how to integrate the left side.

Are you saying you can't work \int {\frac{1}{g- \frac{kv}{m}} }dv ?Could you do something like \int (3 -5v)^{-1} dv ?

 

[Math10]

No, now I got it. I simplified and got gm-kv=Ce^(-kt/m).