What do you do if your exact equation isn't exact? and they give u an Integrating F

by mr_coffee
Tags: equation, exact, integrating
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,568 Do you understand what an "integrating factor" is? If a differential equation is not exact then there always exists some function f(x,y) so that multiplying the equation by it makes it exact. Unfortunately, it's most often very difficult (if not impossible) to find that integrating factor! In this case your equation is x2y3dx+ x(1+ y2)dy= 0. Yes, that is NOT exact because (x2y3)[sub]y[sub]= 3x[sup]2[sup]y2 which is not the same as (x(1+ y2))x= 1+y2. Fortunately, you were told that $\frac{1}{xy^3}$ is an integrating factor. That means that multiplying the equation by that: (1/xy2){x2y3dx+ x(1+y2)dy = ydx+ (1+y2)y2)dy= 0. Is that exact? It certainly should be! If it is exact can you solve it?