# Solve the differential equation F=F0+kv

1. Sep 25, 2016

### OmegaKV

1. The problem statement, all variables and given/known data

Find the velocity of v as a function of displacement x for a particle of mass m which starts from rest at x=0 and subject to the following force:

$$F=F_0+kv$$

You could say mv = F0*t + kx, but the answer in the back of the book is an equation that is only in terms of x and v, not t. The answer in the back of the book involves ln.

2. Relevant equations

Maybe this:

$$\ddot {x}= \frac{d \dot{x}}{dx}$$

3. The attempt at a solution

$$m\ddot{x}=F_0 +k\dot{x}$$
$$m\dot{x} \frac{d\dot{x}}{dx}=F_0 +k\dot{x}$$
$$m\dot{x} d\dot{x} = F_0dx + k\dot{x} dx$$
$$\frac{1}{2}m\dot{x}^2 = F_0*x + ???$$

Last edited: Sep 25, 2016
2. Sep 25, 2016

### Orodruin

Staff Emeritus
The second equation you wrote down in your attempted solution is separable, but in order to separate it the side with dx should not depend on $\dot x$.

3. Sep 27, 2016

### rude man

Is this eq'n dimensionally correct?

4. Sep 27, 2016

### Orodruin

Staff Emeritus
Well, he is not actually using that equation. He is using $\ddot x = \dot x \, d\dot x/dx$.

5. Sep 28, 2016

### rude man

Assuming F0 and k are constants, how about a substitution of variables to reduce the 2nd order linear ODE into a 1st, then taking orodruin's hint to employ separation of variables to solve the new equation?

6. Sep 28, 2016

### rude man

Right, but he wrote it down & should learn to check for dimensional consistency, a powerful error-detecting tool that doesn't seem to be sufficiently emphasized in our classrooms.