Closed form of a differential equation (easy one)

MonicaCFA · September 6th, 2006, 11:25 pm

Hello Financial Studs, can anyone give me a hand solving the following?Show thatcan be solved in closed form, namelyI know that I have to use partial fractions, that is, How can I get C1 and C2, and then how can I integrate the resulting term Thanks in advance for any hint!

kanukatchit · September 7th, 2006, 12:34 am

Well.. arent you a darling..if its easy why ask... homework is it ?here is a hintBy the way you have a differential equation dx/dt = e x - sig x^2with boundary x(0) = Xyour partial fraction numerator is xC1 + (e - sigx) C2 = 1x(C1 - sigC2) + eC2 = 1LHS has x and a const term. RHS has only const term.Therefore coef of x has to be zero and..C1 - sigC2 = 0eC2 = 1-> C2 = 1/esolve for C1. only thing left is integration..Edit : negative sign missed

Jungix · September 7th, 2006, 11:35 am

That's what I would have done if I had been given only the ODE.Since you're given the solution you can be lazy and just check that it is indeed a solution. That should do it for your homework

MonicaCFA · September 7th, 2006, 8:38 pm

You guys are fascinatingly smart. Hints were very helpful, I was getting a wrong C1 because of algebra error.Thanks!