Can't figure out how to solve a diffrential equation

[johnt447]

Hey

Not done this in ages and just can't figure it out, i need to solve the equation;

$\frac{dv}{dt}=-\alpha v+\lambda F$

Where alpha,lambda and F are constants.

I'm so used to solving differential equations numerically i think I have forgotten how to do it analytically:P

My first attempt was looking at the integrating factor method although that seemed to get a bit messy think i might have messed it up a bit.

 

[bpatrick]

ok to continue with your integrating coefficient method ... a good start would be to use e^αt and multiply that through the entire equation after you move the αv over to the left side.

dvdt=−αv+λFdvdt+αv=λF

e^αt dvdt + αv e^αt= λF e^αt

the left side is the derivative (chain/product rule) of some function: ddt (v e^αt) ... (since v is a function of t the chain + product rules both happen.) so if you rearrange the left side a bit and express it as a derivative like I said up there, then integrate both sides with respect to t, you'll have your solution.

I'm not going to finish the integration and algebra just due to whatever academic integrity forum rules there probably are, but you should get something in the general form of:

v = [v(0) - λF/α] e^-αt + λF/α

for v=v(0) at t = 0

... assuming I didn't make any silly mistakes or mistypes on my phone.

I hope that kinda helps more than it confuses

 

[dacruick]

This is the method that you need to use.

http://mathworld.wolfram.com/First-OrderOrdinaryDifferentialEquation.html

You can also use this website to compute DEs

http://www.wolframalpha.com/input/?i=y''+++y+=+0

It might not show you the work, but you can check your answer using it.

 

[johnt447]

Thanks for the replies and that webstie to check DE is really useful.

 

[dacruick]

Thanks for the replies and that webstie to check DE is really useful.

Yes it's a really cool tool for graphing and that kind of thing too. It pretty much does everything that you can think of, and usually shows its method.