Question

Find an integrating factor to make the ODE exact:

$(x^{2} - y^{2} - y)dx - (x^{2} - y^{2} - x)dy = 0$

Answer 1

solution- The differential equation of the form - MỨC, 4) dọc + N2, 4) du =ó is exact differential equation if it satisfies the condition The given differenticel equation is: (ac2 ye-y) dx - (02-y²-x) dy = o al compare equcation (1) with Muay) dx + Ncac yn we get T = x²-y²y & N=- (x2-y²-x) A = - 2x + 1 by - OM ON OX e Differential equation (1) is Non-Exact. Integrating factor - Seppose there exists a function fox,y) such that, - f(x,y) [Moxy) dx + NCR y) dy] =0 is exact. Then, fxy) is called as an Integrating factor from equation a), we can write, (x2-yl) dx- y dx - exeye dy + x dy =oLaw Multiply the above equation by f(x,y) = Luer

we get, dx - e lye dx - dy + x dy = 0. > { yde dx + (a1 + y ) dycony - Again compare Mocy) dx + M= - Y xeye this equation with Ncx.y dy=o, we get & N= - 1 + x -By - o - foto ngentot cey)] & JM = IX = ot Cocl-y) (1) -x (2x) (22 y ² 2 x - y ² - 201² Cx2ye) e - 42-X - = (xc2ye, 2 (x²-4872

JM - by DN ax . Equation (Ill) is Exact differential equation and its integrating factor. IcF = f(x, y) = y bue