My Math Forum Differential equation problem

 Differential Equations Ordinary and Partial Differential Equations Math Forum

 June 27th, 2018, 10:24 AM #1 Newbie   Joined: Jun 2018 From: Netherlands Posts: 2 Thanks: 1 Differential equation problem Hi guys, I have a (small) problem with a differential equation I want to solve. Now I know the answer, but I do not know how to get there, which is essential. The differential equation is the following: dm/dp = a+b*p+c*m The answer is: m=u*exp(c*p)-(1/c)*(b*p+b/c+c) where u is the constant of integration. Can somebody please explain the steps to me? Many thanks in advance!!
 June 27th, 2018, 10:46 AM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,204 Thanks: 511 Math Focus: Calculus/ODEs We are given to solve: $\displaystyle \frac{dm}{dp}=a+bp+cm$ I would first arrange in standard linear form: $\displaystyle \frac{dm}{dp}-cm=a+bp$ Compute the integrating factor: $\displaystyle \mu(p)=\exp\left(-c\int \,dp\right)=e^{-cp}$ Multiply the ODE in standard form by this factor: $\displaystyle e^{-cp}\frac{dm}{dp}-ce^{-cp}m=(a+bp)e^{-cp}$ Rewrite the LHS: $\displaystyle \frac{d}{dp}\left(e^{-cp}m(p)\right)=(a+bp)e^{-cp}$ Integrate w.r.t $\displaystyle p$: $\displaystyle e^{-cp}m(p)= -\frac{e^{-cp}(ac+bcp+b)}{c^2}+C$ Hence: $\displaystyle m(p)=Ce^{cp}-\frac{ac+bcp+b}{c^2}$ This is almost the result you posted as the solution...did you copy it correctly? Thanks from econstudent1993
 June 27th, 2018, 10:57 AM #3 Newbie   Joined: Jun 2018 From: Netherlands Posts: 2 Thanks: 1 Thank you so much! You're right, I made a small typo in the answer. The last c should be an a. Thanks from MarkFL
 June 27th, 2018, 12:02 PM #4 Global Moderator   Joined: Dec 2006 Posts: 19,522 Thanks: 1747 I'll assume that c is not zero. The equation implies e^(-cp)dm/dp - cme^(-cp) = ae^(-cp) + bpe^(-cp). Integrating w.r.t. p gives e^(-cp)m = -(a/c)e^(-cp) - (b/c)pe^(-cp) - (b/c²)e^(-cp) + u, where u is a constant. Multiplying by e^(cp) gives m = ue^(cp) - (b/c)p - b/c² - a/c, which is slightly different from what you gave as the answer. If c is zero, what solution do you get?

 Tags differential, equation, problem

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post BonaviaFx Calculus 6 July 19th, 2015 08:45 AM Zoser Calculus 17 February 20th, 2015 02:44 PM ricsi046 Calculus 8 May 30th, 2014 11:33 AM greg1313 Differential Equations 11 July 11th, 2011 11:38 PM sivela Differential Equations 1 January 21st, 2011 05:53 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top