How do I solve a separable differential equation? Can you give a worked example?

For dy/dx = y*x, separate variables: dy/y = x dx. Integrate both sides: ln|y| = x^2/2 + C. Exponentiate: |y| = e^(x^2/2 + C) = Ae^(x^2/2) where A = e^C. General solution: y = Ae^(x^2/2). Always add constant C on ONE side only. For initial value problems, substitute the given point to find A. Common mistake: forgetting the absolute value signs and the constant of integration. Key step: always rearrange so all y terms and dy are on one side, all x terms and dx are on the other side, THEN integrate.