The give differential equation is-
(x + 2y − 1)dx − (2x + y − 5)dy = 0
first, re-arrange the problem to be-
x+2y−1+(2x+y−5)dxdy=0
let M=x+2y−1 and N=2x+y−5 and define Fx=M and Fy=N
Differentiate M w.r.t y-
My=2
Differentiate N w.r.t. x-
Nx=2
As My=Nx
thus the problem is an EXACT ODE
fourth, find integral of M dx holding y constant =2x2+2xy−x+h(y)=F
find Fy=2x+h′=N=2x+y−5... ' solve for h′=y−5 and h=2y2−5y
Complete solution is -
2x2+2xy−x+2y2−5y=c
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