Separable ODE Guide
Solve first-order separable differential equations step by step, with the initial condition built in.
Separable means you can get all y’s on one side and all x’s on the other — then integrate both sides.
① Answer
② Steps
③ Why it works
A separable equation dy/dx = g(x)·h(y) splits into (1/h(y))dy = g(x)dx — the variables no longer mix. Integrating each side separately gives an implicit solution; the initial condition then pins down the constant C. Exponential growth (dy/dx = ky) is the flagship: its solution y = y₀e^{kx} appears in population models, cooling, and compound interest.