Answer to Question #190509 in Differential Equations for randz

Question #190509

5. (x + 2y − 1)dx − (2x + y − 5)dy = 0


1
Expert's answer
2021-05-11T15:05:05-0400

The give differential equation is-

(x + 2y − 1)dx − (2x + y − 5)dy = 0


first, re-arrange the problem to be-


x+2y1+(2x+y5)dydx=0x+2y-1+(2x+y-5)\dfrac{dy}{dx}=0


let M=x+2y1M=x+2y-1 and N=2x+y5N=2x+y-5 and define Fx=MFx=M and Fy=NFy=N


Differentiate M w.r.t y-

My=2M_y=2


Differentiate N w.r.t. x-

Nx=2N_x=2


As My=NxM_y=N_x

thus the problem is an EXACT ODE


fourth, find integral of M dx holding y constant =x22+2xyx+h(y)=F= \dfrac{x^2}{2}+2xy-x+h(y) = F


find Fy=2x+h=N=2x+y5...F_y= 2x+h'= N = 2x+y-5 ... ' solve for h=y5h'=y-5 and h=y225yh=\dfrac{y^2}{2}-5y


Complete solution is -


x22+2xyx+y225y=c\dfrac{x^2}{2}+2xy-x+\dfrac{y^2}{2}-5y=c


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