AHL 5.14 (HL)—Differential equation modelling

Build a differential equation by translating the stated rate relationship and units. A separable equation $dy/dx=g(x)h(y)$ can be rearranged as $dy/h(y)=g(x)\,dx$ and integrated on both sides.

The integration constant gives the general solution; an initial condition selects a particular solution. For proportional change $dy/dx=ky$, separation gives $\ln|y|=kx+C$ and hence $y=Ae^{kx}$ on the relevant branch.

Example

If $dP/dt=0.2P$ and $P(0)=100$, then the general non-zero solution is $P=Ae^{0.2t}$ and the condition gives $A=100$, so $P(t)=100e^{0.2t}$.

Do not introduce a logistic capacity term unless the context states it. Keep equilibrium solutions that may be lost by division, and distinguish a general solution from the one satisfying initial data.