Solving dy/dx = y²x

Differential Equations · Axiom Academy

Work through the separation-of-variables technique step by step on a real problem. Solve the differential equation using separation of variables. Nice work — you've solved a separable differential equation from start to finish. Here's what carried the solution: Spot separable form: factors as a function of y times a function of x , so it's separable. Separate algebraically: divide by y^2 and multiply by dx to get all y -terms (with dy ) on one side and all x -terms (with dx ) on the other. Integrate both sides: and — don't drop the constant of integration. Solve for y : take reciprocals of to get the explicit general solution . Watch for lost solutions: dividing by y^2 assumes — y = 0 is also a solution of the original equation and should be checked separately. Separation of variables is one of the most useful techniques for first-order differential equations — the key skill is recognizing when an equation factors into g(x)h(y) and confidently carrying out the algebra that follows.

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