# Solve the differential equation F=F0+kv

## Homework Statement

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.

## Homework Equations

Maybe this:

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

## The Attempt at a Solution

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

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Orodruin
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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##.

• OmegaKV
rude man
Homework Helper
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## Homework Statement

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.

## Homework Equations

Maybe this:
$$\ddot {x}= \frac{d \dot{x}}{dx}$$
Is this eq'n dimensionally correct?

## The Attempt at a Solution

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

Orodruin
Staff Emeritus
Homework Helper
Gold Member
Is this eq'n dimensionally correct?
Well, he is not actually using that equation. He is using ##\ddot x = \dot x \, d\dot x/dx##.

rude man
Homework Helper
Gold Member

## Homework Statement

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:
$$F=F_0+kv$$

## The Attempt at a Solution

$$m\ddot{x}=F_0 +k\dot{x}$$
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?

rude man
Homework Helper
Gold Member
Well, he is not actually using that equation. He is using ##\ddot x = \dot x \, d\dot x/dx##.
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.