How a Cooling Cup of Chai Obeys a Differential Equation

A hot cup of chai cools faster when it is much hotter than the room, and a bank balance grows faster when it is larger. Both describe a rate of change in terms of the quantity itself — a differential equation. Solving it tells us the quantity at any moment.

This part covers order and degree, general and particular solutions, and solving by separation of variables.

**The order is the highest derivative present, and the degree is the power of that highest derivative once the equation is a polynomial in its derivatives; if a derivative appears inside a function such as $\sin$ or $\log$, the degree is not defined.

Definitions:

- Differential equation — an equation involving derivatives of a dependent variable
- Order — the order of the highest derivative in the equation
- Degree — the highest power of the highest-order derivative, when the equation is a polynomial in its derivatives

Worked examples:**

- $\frac{dy}{dx}=\cos x$ — order $1$, degree $1$
- ${\left(\frac{d^{2}y}{d{x}^2}\right)}^3+{\left(\frac{dy}{dx}\right)}^2+y=0$ — order $2$, degree $3$
- $\frac{d^{3}y}{d{x}^3}+2{\left(\frac{d^{2}y}{d{x}^2}\right)}^2-\frac{dy}{dx}+y=0$ — order $3$, degree $1$
- $\frac{d^{2}y}{d{x}^2}+\sin \left(\frac{dy}{dx}\right)=0$ — order $2$, degree not defined

Order is always defined, and order and degree, when defined, are positive integers.

An everyday example. A rule about how fast a car moves involves the first derivative of its position, while a rule about how sharply it brakes involves the second — equations of order $1$ and order $2$.

The substance. Clear radicals before reading the degree — for $\sqrt{1+(y^{\prime }{)}^2}=y^{\prime \prime }$, squaring gives $1+(y^{\prime }{)}^2=(y^{\prime \prime }{)}^2$, so the degree is $2$.

A general solution contains as many arbitrary constants as the order of the equation, a particular solution gives those constants definite values, and a function is verified as a solution by substituting it and its derivatives into the equation.

Two kinds of solution:

- General solution — contains arbitrary constants, such as $y=a\cos x+b\sin x$
- Particular solution — free of arbitrary constants, such as $y=3\cos x$

Worked example 1 — verify. Show that $y=e^{-3x}$ satisfies $\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}-6y=0$.

$$y^{\prime }=-3e^{-3x},y^{\prime \prime }=9e^{-3x}$$

$$y^{\prime \prime }+y^{\prime }-6y=9e^{-3x}-3e^{-3x}-6e^{-3x}=0$$

Worked example 2 — a general solution. $y=a\cos x+b\sin x$ satisfies $y^{\prime \prime }+y=0$ for all values of $a$ and $b$, since $y^{\prime \prime }=-a\cos x-b\sin x=-y$. It has two constants, matching the order $2$.

An everyday example. Every ball thrown straight up follows the same law of motion, but its exact flight depends on the speed and height it starts from — a general solution becomes particular once those starting values are known.

The substance. The number of arbitrary constants equals the order — a second order equation needs two conditions to fix a particular solution.

**If the equation can be written as $\frac{dy}{dx}=g(x)h(y)$, rearrange it to $\frac{dy}{h(y)}=g(x)dx$, integrate both sides, add one constant, and use any initial condition to find that constant.

Worked example 1.** Solve $\frac{dy}{dx}=\frac{x+1}{2-y}$, with $y\ne 2$.

$$(2-y)dy=(x+1)dx\Rightarrow 2y-\frac{y^2}{2}=\frac{x^2}{2}+x+C$$

Worked example 2 — with an initial condition. Solve $\frac{dy}{dx}=2xy$, given $y=3$ when $x=0$.

$$\frac{dy}{y}=2xdx\Rightarrow \log |y|=x^2+C$$

At $x=0$, $\log 3=C$, so $\log |y|=x^2+\log 3$, that is,

$$y=3e^{x^2}$$

Worked example 3 — growth. Bacteria multiply at a rate proportional to their number, $\frac{dP}{dt}=kP$. Separating gives $P=P_0{e}^{kt}$. With $P_0=1000$ and $k=0.1$ per hour, after $10$ hours

$$P=1000e^1\approx 2718$$

An everyday example. A cup of hot chai on the table cools according to a separable equation: its rate of cooling is proportional to how much warmer it is than the room.

The substance. Only one constant is needed — constants added to both sides simply combine into one.