Differential Equations — chapter notes

A differential equation is an equation involving an unknown function and its derivatives. When the unknown $y$ depends on one variable $x$ it is an ordinary differential equation, e.g. $\dfrac{dy}{dx} = 3x^2$, $\dfrac{d^2y}{dx^2} + y = 0$.

The order is the order of the highest derivative present. Powers do not affect the order: $\dfrac{d^3y}{dx^3} - \left(\dfrac{dy}{dx}\right)^4 = x$ has order $3$.

If a differential equation is a polynomial in its derivatives, its degree is the highest power (a positive integer) of the highest-order derivative.

• $\left(\dfrac{d^2y}{dx^2}\right)^2 + 3x\left(\dfrac{dy}{dx}\right)^3 = \sin x$: order $2$, degree $2$.
• $\dfrac{d^3y}{dx^3} + \log\left(\dfrac{dy}{dx}\right) = 0$: order $3$, degree not defined.

Functions of $x$ alone ($\sin x$, $e^{x}$, $\log x$) do not stop the degree from being defined; a derivative inside such a function does.

A solution is a function that satisfies the equation. The general solution contains arbitrary constants, as many (independent) as the order. A particular solution has all constants fixed, usually by an initial condition like $y(x_0) = y_0$.

To verify that a given function is a solution: differentiate it, substitute into the equation, and show that both sides agree. Example: $y = e^{-x} + 2x - 2$ satisfies $\dfrac{dy}{dx} + y = 2x$.

Not in the 2026–27 syllabus: forming a differential equation from a given family by eliminating arbitrary constants.

If $\dfrac{dy}{dx} = g(x)\,h(y)$, separate and integrate:

Examples: $\dfrac{dy}{dx} = x(1 + y^2)$ gives $\tan^{-1}y = \dfrac{x^2}{2} + C$; $\dfrac{dy}{dx} = y\cos x$ with $y(0) = 2$ gives $y = 2e^{\sin x}$.

Growth and decay: $\dfrac{dN}{dt} = kN$ gives $N = N_0e^{kt}$. Equal time intervals multiply $N$ by equal factors.

$F(x, y)$ is homogeneous of degree $n$ if $F(\lambda x, \lambda y) = \lambda^nF(x, y)$. The equation $\dfrac{dy}{dx} = F(x, y)$ is homogeneous when $F$ has degree $0$.

Method: put $y = vx$, so $\dfrac{dy}{dx} = v + x\dfrac{dv}{dx}$; simplify to $x\dfrac{dv}{dx} = g(v)$; separate $\dfrac{dv}{g(v)} = \dfrac{dx}{x}$; integrate; replace $v$ by $\dfrac{y}{x}$.

Example: $xy\dfrac{dy}{dx} = x^2 + 2y^2$ gives $x^2 + y^2 = Ax^4$; with $y(1) = 1$, $A = 2$.

(For $\dfrac{dx}{dy} = G(x, y)$ with $G$ of degree $0$, the substitution $x = vy$ works the same way.)

Standard form: $\dfrac{dy}{dx} + Py = Q$, where $P$ and $Q$ are functions of $x$ (or constants).

Integrating factor: $\text{I.F.} = e^{\int P\,dx}$ (no constant needed).

Solution: $y \cdot \text{I.F.} = \int Q \cdot \text{I.F.}\,dx + C$.

Examples: $\dfrac{dy}{dx} + 3y = e^{-x}$ gives $y = \dfrac{e^{-x}}{2} + Ce^{-3x}$; $\cos x\dfrac{dy}{dx} + y\sin x = \cos^2 x$ (divide by $\cos x$ first) gives $y = (x + C)\cos x$.

Linear in $x$: $\dfrac{dx}{dy} + Px = Q$ with $P$, $Q$ functions of $y$: $\text{I.F.} = e^{\int P\,dy}$ and $x \cdot \text{I.F.} = \int Q \cdot \text{I.F.}\,dy + C$. Example: $\dfrac{dx}{dy} + x = y + 1$ gives $x = y + Ce^{-y}$.

Choosing a method: separable? homogeneous of degree $0$? linear in $y$ or in $x$? An equation may fit more than one type.

Applications: cooling $\dfrac{dT}{dt} = -k(T - T_0)$ gives $T = T_0 + (T(0) - T_0)e^{-kt}$; a constant input with loss, $\dfrac{dC}{dt} + aC = b$, approaches the steady value $\dfrac{b}{a}$.