Revision sheet

Differential equations

• Check a solution: compute its derivatives, substitute, compare.
• Equilibrium: set the rate equal to 0.

Watch out: equilibrium means $y' = 0$, not $y = 0$.

Slope fields

• Slope at $(x, y)$: substitute into $f(x, y)$.
• Only $x$: parallel along verticals. Only $y$: along horizontals.
• $f = 0$: horizontal segments (equilibria if constant in $y$).

Watch out: $(x, y)$ in that order.

Separation of variables

1. $\dfrac{dy}{h(y)} = g(x)\,dx$.
2. Integrate both sides, one $+C$.
3. Solve for $y$; use the initial value.

Watch out: $e^{x + C} = Ae^x$, not $e^x + C$.

Exponential models

• $y' = ky$, $y(0) = y_0 \Rightarrow y = y_0e^{kt}$.
• Decay: $k < 0$; $k = -\ln 2 / (\text{half-life})$.

Watch out: equal times multiply by equal factors.

Newton's law of cooling

• $T' = -k(T - A)$.
• $T = A + (T_0 - A)e^{-kt}$, approaching $A$.

Watch out: it is the difference $T - A$ that decays.