An introduction to differential equations
Get started with differential equations for Year 12 Specialist Mathematics in Queensland (QCAA). A differential equation links an unknown function to its derivatives, and is the natural way to describe growth, decay, cooling and motion — the foundation for the rest of the rates of change topic.
You will learn to read off the order of a differential equation, verify that a given function is a solution by substitution, tell a general solution from a particular one, and form a differential equation from a described rate such as Newton's law of cooling or radioactive decay.
Theory
A differential equation is an equation that links an unknown function to its derivatives. In Year 12 Specialist Mathematics (QCAA, Queensland) they are the natural language of change — growth, decay, cooling and motion. This page explains what a differential equation is, its order, how to verify a given function is a solution, the difference between general and particular solutions, and how to form a differential equation from a described rate.
A differential equation (a "DE") is an equation that contains one or more derivatives of an unknown function, for example \(\dfrac{dy}{dx}=2y\) or \(\dfrac{d^2y}{dx^2}=-9y\). It describes how a quantity changes, rather than giving the quantity directly. An equation with no derivative, such as \(y=2x+1\) or \(x^2+y^2=25\), is not a differential equation.
The order of a differential equation is the order of the highest derivative that appears. So \(\dfrac{dy}{dx}=3x^2-5\) is first order, while \(\dfrac{d^2y}{dx^2}+3\dfrac{dy}{dx}=0\) is second order. A power of a derivative does not raise the order: in \(\dfrac{d^2y}{dx^2}+\left(\dfrac{dy}{dx}\right)^3=\sin x\) the highest derivative is still the second, so it is second order.
A function is a solution of a differential equation if substituting the function and its derivatives makes the equation true for all \(x\). You verify a proposed solution by differentiating it, substituting into both sides, and checking they agree.
The general solution contains arbitrary constants (one for each order — each integration adds one), so it represents a whole family of curves. A particular solution uses an initial or boundary condition (such as \(y(1)=4\)) to fix those constants and pick out one member of the family.
A first-order differential equation relates \(\dfrac{dy}{dx}\) to \(x\) and \(y\). The Year 12 course meets three standard shapes:
A rate proportional to the amount present gives the growth / decay equation, whose general solution is an exponential:
Newton's law of cooling (rate proportional to the gap above the surrounding temperature \(T_s\)) and radioactive decay are the standard rate models:
How to work with a differential equation
- Read off the order: find the highest derivative that appears — that number is the order (a power of a derivative does not change it).
- Verify a solution: differentiate the proposed function as many times as needed, substitute it and its derivatives into both sides, and check the equation holds.
- General then particular: the general solution carries one arbitrary constant per order; substitute the initial or boundary condition to solve for the constant(s).
- Form a DE from a rate: write the rate as a derivative, translate "proportional to" as "\(=k\times\)", and put a minus sign in front when the quantity is decreasing.
The order is the order of the highest derivative, not the highest power. The fourth power sits on a first derivative, so it does not raise the order:
| \(\left(\dfrac{dy}{dx}\right)^4\) | \(\rightarrow\) | \(\text{power of a 1st derivative}\) |
| \(\dfrac{d^2y}{dx^2}\) | \(\rightarrow\) | \(\text{a 2nd derivative}\) |
| \(\text{highest derivative}\) | \(=\) | \(\text{2nd}\) |
| \(\text{order}\) | \(=\) | \(2\) |
The differential equation is second order.
Differentiate \(y=\cos 3x\) twice to build the left-hand side:
| \(y\) | \(=\) | \(\cos 3x\) |
| \(\dfrac{dy}{dx}\) | \(=\) | \(-3\sin 3x\) |
| \(\dfrac{d^2y}{dx^2}\) | \(=\) | \(-9\cos 3x\) |
Substitute into both sides of the differential equation and compare:
| \(\text{LHS}=\dfrac{d^2y}{dx^2}\) | \(=\) | \(-9\cos 3x\) |
| \(\text{RHS}=-9y\) | \(=\) | \(-9\cos 3x\) |
| \(\text{LHS}\) | \(=\) | \(\text{RHS} \ \checkmark\) |
Both sides equal \(-9\cos 3x\), so \(y=\cos 3x\) is a solution.
Substitute the condition \(y(1)=5\) into the general solution to fix \(c\):
| \(y\) | \(=\) | \(x^3+c\) |
| \(y(1)\) | \(=\) | \(1^3+c\) |
| \(5\) | \(=\) | \(1+c\) |
| \(c\) | \(=\) | \(5-1\) |
| \(c\) | \(=\) | \(4\) |
Write the particular solution with this value of \(c\):
| \(y\) | \(=\) | \(x^3+4\) |
The particular solution is \(y=x^3+4\).
Write the rate as a derivative, and the difference from the room as \(T-25\):
| \(\text{rate of change}\) | \(=\) | \(\dfrac{dT}{dt}\) |
| \(\text{gap above the room}\) | \(=\) | \(T-25\) |
The tea is cooling, so the rate is negative; "proportional to" gives a constant \(k>0\):
| \(\dfrac{dT}{dt}\) | \(\propto\) | \(-(T-25)\) |
| \(\dfrac{dT}{dt}\) | \(=\) | \(-k(T-25)\) |
The differential equation is \(\dfrac{dT}{dt}=-k(T-25)\).
Common pitfalls
Frequently asked questions
What is a differential equation?
It is an equation that contains one or more derivatives of an unknown function, such as \(\dfrac{dy}{dx}=2y\). It describes how a quantity changes rather than giving the quantity directly.
What does the order of a differential equation mean?
The order is the order of the highest derivative that appears. \(\dfrac{dy}{dx}=4x\) is first order and \(\dfrac{d^2y}{dx^2}+3\dfrac{dy}{dx}=0\) is second order. A power of a derivative does not raise the order.
How do I verify that a function is a solution?
Differentiate the function as many times as the equation needs, substitute it and its derivatives into both sides, and check the two sides are equal for all \(x\).
What is the difference between a general and a particular solution?
The general solution contains arbitrary constants (one per order) and represents a family of curves. A particular solution uses an initial or boundary condition to fix those constants and select one curve.
How many arbitrary constants does the general solution have?
Exactly as many as the order of the differential equation, because each integration introduces one constant. A third-order equation therefore has three arbitrary constants.
How do I form a differential equation from a described rate?
Write the rate of change as a derivative, replace "proportional to" with "equals \(k\) times", and include a minus sign when the quantity is decreasing, for example \(\dfrac{dP}{dt}=kP\) for growth or \(\dfrac{dN}{dt}=-kN\) for decay.