(4x+3y^2)dx+2xy*dy=0 by using integrating fector

The ODE is exact if .




This is not the case, so look for an integrating factor such that



Since is a function of only, you have



So, the integrating factor is



Now the ODE can be modified as



Check for exactness:




so the modified ODE is indeed exact.

Now, you're looking for a solution of the form , since differentiating via the chain rule yields



Matching up components, you would have





Differentiate this with respect to to get




So the solution here is



Just for a final check, take the derivative to get back the original ODE:






so the solution is correct.