The Constant of Integration

[fizzacist]

While taking an AP physics practice exam, I encountered a difference in the way I solve a differential equation and the way the exam's rubric solves it.

The equation is as follows:

$\frac{dv}{dt}$ = $\frac{F-KV}{m}$

My solution:

$\int$$\frac{dv}{F-KV}$ = $\int$ $\frac{dt}{m}$

u = F-KV

$\frac{du}{-K}$ = dv

$\frac{-1}{K}$ $\int$$\frac{1}{u}$du = $\int$$\frac{dt}{m}$

Integrate that to find

ln|F-KV|+C = -K$\frac{t}{m}$

But before I go any further, the 1993 Exam's Rubric shows that by integrating $\int$$\frac{dv}{F-KV}$ should yield ln|F-KV|-lnC

To me, this makes no sense. The constant of integration should be ln|u| + C, not ln|u|-lnC

Here's what I'm talking about:
http://imgur.com/c83p1
I've also attached the '93's rubric to this post. The problem I'm referring to is problem #2.

Can any of the math/physics gurus out there help me out? :P
Thanks

 

[Hurkyl]

To me, this makes no sense. The constant of integration should be ln|u| + C, not ln|u|-lnC

What's the difference?

Maybe you're confused because you're using C for two different things. Try comparing

• ln|u| + D, and
• ln|u|-lnC

 

[mathman]

Call your constant of integration K (remember K is completely arbitrary). Now define another constant C by K = -lnC. This gives the formula in the book. Since C is also completely arbitrary, it doesn't matter.

 

[fizzacist]

Ahh. Got it. :P

 

[PJC01]

for bringing this issue to my attention! I can understand your confusion about the constant of integration in this particular problem. The constant of integration is a fundamental concept in calculus and it plays an important role in solving differential equations. In this case, the constant of integration represents the unknown initial condition of the system.

It is important to note that the constant of integration can take on different forms depending on the method of integration used. In your solution, you used the substitution method to solve the differential equation, which resulted in ln|F-KV|+C. However, the rubric for the 1993 exam used the method of partial fractions, which resulted in ln|F-KV|-lnC.

Both solutions are mathematically correct and yield the same general solution to the differential equation. The difference lies in the interpretation of the constant of integration. In your solution, the constant of integration represents the initial condition of the system, while in the rubric's solution, the constant of integration represents the integration constant that arises from using partial fractions.

In conclusion, there is no need to be concerned about this difference in notation. As long as the solution is mathematically correct and yields the same general solution, it is acceptable. The important thing is to understand the concept of the constant of integration and how it relates to the initial condition of the system in your particular problem. Keep up the good work in your AP physics studies!