Solving (2xy + 3)dx + (x² - 1)dy = 0

Differential Equations · Axiom Academy

Verify exactness, find the potential function through partial integration, and determine the implicit solution. Solve the differential equation by verifying it is exact and finding the potential function F(x,y) . Excellent work! You've successfully solved an exact differential equation. Here's what we learned: Exactness test: A differential equation is exact if and only if . This is the crucial first step. Finding the potential function: Integrate M with respect to x to get , where h(y) is an unknown function of y alone. Determining h(y) : Take , set it equal to N(x,y) , and solve for h'(y) . Then integrate to find h(y) . Implicit solution: The solution is F(x,y) = C , where C is an arbitrary constant — this represents a family of solution curves. Verification: Always double-check your partial derivatives at each step to ensure accuracy. This systematic approach works for every exact differential equation. When an equation isn't exact, an integrating factor can sometimes be found to make it exact.

This is the written version of the interactive lesson above. See the full Differential Equations course.