# How was this ODE solution found? Doesn't seem to be the normal solution.

• InvisibleMan1
In summary, the ODE solution was found through a series of mathematical calculations and techniques, such as separation of variables and substitution. It may not seem like the conventional approach, but it is a valid and accurate method for solving differential equations.
InvisibleMan1

## Homework Statement

The solution for the differential equation on this page http://electron9.phys.utk.edu/phys135d/modules/m5/Friction.htm#Drag checks out, but I can't figure out how they found it. Both my solution and theirs check out. A couple people I asked for help reached the same solution I did.

The problem appears to be with the constant of integration. I don't see a way of getting from mine to theirs though.
What I am guessing theirs is: C = -((kv(0)/g + 1)
Mine: C = (kv(0))/g - 1

I'm starting to suspect that their solution is a very lucky typo. "Lucky" since it is actually a valid solution, unless I checked it wrong.

## Homework Equations

The differential equation: dv/dt = g - (b/m)v

Their solution: v = (g/k) - [(kv(0) + g)/k]e^(-kt) Where k = b/m

The solution I found: v = (g/k) + [(kv(0) - g)/k]e^(-kt) Where k = b/m
The integrating factor I found: u = e^(kt)
The constant of integration I found: C = (kv(0))/g - 1

## The Attempt at a Solution

Write the ODE in standard linear form, and solve it as a first-order linear ODE using an integrating factor u(x)=e^(int(p(x)dx). The solution I and a couple other people reached is given above.

On a side-note, I also can't figure out why they multiplied (kv(0))/g in C by k/k. I did the same thing in my solution just because I couldn't see a reason why not (and it makes it easier to compare the two solutions). I suspect the way they found their solution may have had something to do with k/k, but it is also possible they just did that to make it easier to do/show something.

Last edited by a moderator:
For their solution

$$v(0) = -v_0.$$

They probably have a sign wrong. The only other possibility is that they're trying to choose a convention where the velocity is negative for a falling object, but that doesn't seem to be compatible with the other terms in their solution.

Is my theory about it being a lucky typo correct then?

InvisibleMan1 said:
Is my theory about it being a lucky typo correct then?

I obtain the same solution as you did. Since you wrote v(0) rather than v0 there's no room for confusion.

Erm, sorry, but I don't see how that answers my question.

InvisibleMan1 said:
Erm, sorry, but I don't see how that answers my question.

Their solution is a solution to the DE, but the notation doesn't correspond well to the initial value condition, since

$$v_0 = - v(0).$$

The minus sign could very well be a typo, the only time they refer to v0 is to say that it's the speed at t=0 and the sign didn't matter for that.

You got the right solution and handled the initial value condition the right way.

Alright, thanks for the help.

## FAQ: How was this ODE solution found? Doesn't seem to be the normal solution.

The ODE solution was likely found using a mathematical method, such as separation of variables or substitution, in order to solve the differential equation and find the function that satisfies it.

## 2. Why does the solution not match what I expected?

The solution to an ODE can take many different forms depending on the specific equation and initial conditions. It is important to carefully check the steps used to obtain the solution to ensure accuracy.

## 3. Is this the only solution to the ODE?

In general, a differential equation can have multiple solutions. However, if the equation is well-defined and the initial conditions are specified, there may only be one unique solution.

## 4. How can I verify that this solution is correct?

There are several methods for verifying the correctness of an ODE solution, such as plugging it into the original equation and checking if it satisfies it, or using numerical methods to approximate the solution and comparing it to the analytical one.

## 5. Can this solution be simplified further?

In some cases, the solution to an ODE can be simplified by using trigonometric identities or other mathematical techniques. However, it is important to carefully consider the validity of any simplifications and ensure they do not alter the solution.

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