Turn a rate law into a model

A differential equation states how a quantity changes. For separable equations, collect the dependent variable on one side and the independent variable on the other, integrate, and use an initial condition.

Check constant equilibrium solutions before dividing by a function of the dependent variable. Direction fields show local slopes, while models such as cooling and logistic growth connect the rate to the current state.

\displaystyle \frac{dy}{dx}=f(x)g(y)\Rightarrow\int\frac{dy}{g(y)}=\int f(x)\,dx

where

\displaystyle g(y)\ne0

WORKED EXAMPLE

Solve dy/dt = 2y with $y(0)$ = 3.

Separate: dy/y = 2 dt.
Integrate ln|y| = 2t + C, giving y = Ae²ᵗ.
The initial condition gives A = 3, so y = 3e²ᵗ.

Assumed knowledge

Differentiate and integrate elementary functions, including exponentials and logarithms. Rearrange equations, use initial conditions, and interpret a derivative as a local rate or slope.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Turn a rate law into a model

Solve dy/dt = 2y with y(0)y(0)y(0) = 3.

Put the idea to work.

Solve dy/dt = 2y with $y(0)$ = 3.