A differential equation relates a function to its derivative. Separable ones (dy/dx = f(x)g(y)) are solved by separating variables and integrating both sides. Worked example Separable DE Solve dy/dx = 6x², given y = 5 when x = 0. Integrate both sides: y = ∫ 6x² dx = 2x³ + c. Use the initial condition y(0) = 5: 5 = 2(0) + c ⟹ c = 5. ✓ y = 2x³ + 5. ✓
Rates in context
Growth/decay problems often give dy/dt = ky, whose solution is y = A e^(kt). Exponential growth (k>0) or decay (k<0): link back to the physical situation.