Differential Equations
Differential Equations tests formation, solution and interpretation of first-order models. Expect separable equations, linear first-order equations with integrating factors, and applied growth, decay or cooling models.
Core rules
- Separable: dy/dx=f(x)g(y) gives ∫dy/g(y)=∫f(x)dx, then apply the initial condition.
- Linear: dy/dx+P(x)y=Q(x). The integrating factor is I=e^{∫P dx}; then (Iy)'=IQ.
- Equilibrium solutions occur where dy/dx=0; stability depends on the sign of dy/dx near the equilibrium.
- Always state the domain of the solution and use the initial condition to fix the constant.
Worked example
Solve dy/dx+2y=4, y(0)=1. Here P=2, so I=e^{2x}. Then (ye^{2x})'=4e^{2x}. Integrating gives ye^{2x}=2e^{2x}+C, so y=2+Ce^{-2x}. The initial condition gives 1=2+C, so C=-1. Thus y=2-e^{-2x}.
Traps
Wrong sign in the integrating factor; multiplying only one side by I; losing the constant; ignoring a restricted domain; confusing an equilibrium value with the initial value; failing to interpret the long-term behaviour.