Mathematics - Differential Equations - First-order ODEs - Solving First-Order ODEs

12 steps: (1) A first-order ODE: dy/dx = f(x,y). We want y(x). (2) SEPARABLE: dy/dx = g(x)h(y). Separate: dy/h(y) = g(x)dx. Integrate both sides. (3) Example: dy/dx = xy. → dy/y = xdx. → ln|y| = x²/2+C. → y=Ae^(x²/2). (4) Initial conditions: if y(0)=3, then 3=A·1, so A=3. (5) LINEAR first-order: dy/