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: 12,211 Thanks: 521 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: 20,757 Thanks: 2138 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 Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded 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

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