A first-order differential equation (ODE) relates $y$, $x$, and $dy/dx$:
$$\frac{dy}{dx} = f(x, y)$$
A separable equation has the form $\dfrac{dy}{dx} = g(x)h(y)$.
Method: Separate variables and integrate both sides:
$$\int \frac{dy}{h(y)} = \int g(x)\,dx$$
Example 1: $\dfrac{dy}{dx} = 2xy$, $y(0) = 3$.
$$\int \frac{dy}{y} = \int 2x\,dx \Rightarrow \ln|y| = x^2 + C \Rightarrow y = Ae^{x^2}$$
Apply IC: \$3 = Ae^0 = A$. So $y = 3e^{x^2}$.
Example 2 (population model): $\dfrac{dP}{dt} = kP(1 - P/M)$ (logistic growth).
Partial fractions: $\dfrac{1}{P(1-P/M)} = \dfrac{1}{P} + \dfrac{1/M}{1-P/M}$.
This gives $\ln\left|\dfrac{P}{M-P}\right| = kt + C$, leading to:
$$P(t) = \frac{M}{1 + Ae^{-kt}}$$
Standard form: $\dfrac{dy}{dx} + P(x)y = Q(x)$.
Integrating factor: $\mu(x) = e^{\int P(x)\,dx}$.
Multiply through by $\mu$: $\dfrac{d}{dx}[\mu y] = \mu Q(x)$, then integrate.
Example 3: $\dfrac{dy}{dx} - \dfrac{y}{x} = x^2$.
$P(x) = -1/x$, so $\mu = e^{\int -1/x\,dx} = e^{-\ln x} = 1/x$.
Multiply: $\dfrac{d}{dx}\left[\dfrac{y}{x}\right] = x$.
Integrate: $\dfrac{y}{x} = \dfrac{x^2}{2} + C \Rightarrow y = \dfrac{x^3}{2} + Cx$.
Example 4: $\dfrac{dy}{dx} + 2y = 4$, $y(0) = 1$.
$\mu = e^{2x}$. Then $\dfrac{d}{dx}[e^{2x}y] = 4e^{2x}$.
Integrate: $e^{2x}y = 2e^{2x} + C \Rightarrow y = 2 + Ce^{-2x}$.
IC: \$1 = 2 + C \Rightarrow C = -1$. So $y = 2 - e^{-2x}$.
Newton’s Law of Cooling: $\dfrac{dT}{dt} = -k(T - T_a)$ where $T_a$ is ambient temperature.
Separable: solution is $T(t) = T_a + (T_0 - T_a)e^{-kt}$.
Example 5: An object at $90^\circ$C cools in a $20^\circ$C room. After 10 min it is $60^\circ$C. Find $k$.
$\$60 = 20 + 70e^{-10k} \Rightarrow e^{-10k} = \frac{40}{70} = \frac{4}{7} \Rightarrow k = -\frac{1}{10}\ln\frac{4}{7} = \frac{\ln(7/4)}{10}$$
Radioactive decay: $\dfrac{dN}{dt} = -\lambda N \Rightarrow N(t) = N_0 e^{-\lambda t}$.
Growth/decay with constant input: $\dfrac{dy}{dt} = a - by$ (linear ODE).
KEY TAKEAWAY: Separable equations use direct separation and integration; linear equations use an integrating factor. The key skill is recognising which method applies.
EXAM TIP: Always include the constant of integration $C$ when solving the ODE, and apply initial conditions afterwards to find $C$. Never substitute the IC before integrating.
COMMON MISTAKE: For separable equations, dividing by $h(y) = 0$ without noting this as a special (constant) solution, or omitting $|y|$ in $\ln|y|$ before exponentiating.
APPLICATION: Differential equation models appear in biology (population growth), physics (cooling, radioactive decay), chemistry (reaction rates), and finance (continuous compounding).
