AHL 5.18 (HL)—Differential equations

A differential equation becomes a model after a condition selects a solution.

The equation states a rate relationship; solving it gives a family, and an initial or boundary condition chooses the member that fits the system.

Example

dy/dx=2y with y(0)=3 gives y=3e^(2x), not the whole family Ce^(2x).

Separate variables or use the appropriate method, then substitute the condition and check the derivative.

A mathematical solution can still be physically invalid if it violates domain or sign constraints.

Method map: separable equations rearrange to $g(y)dy=f(x)dx$; homogeneous $dy/dx=F(y/x)$ uses $y=vx$; linear $y'+P(x)y=Q(x)$ uses integrating factor $\mu=e^{\int P(x)dx}$ so $(\mu y)'=\mu Q$. Euler's method for $y'=f(x,y)$ uses $x_{n+1}=x_n+h$ and $y_{n+1}=y_n+h f(x_n,y_n)$. Apply the initial condition after integration and check the solution in the original equation.