Year 12 Maths Extension 1 (2027) Further applications of calculus

Differential equations – introduction & order

Q: What is the order of a differential equation?

The order of the highest derivative that appears; degree is the power of that derivative.

Q: What is an initial value problem?

A differential equation together with an initial condition that fixes the constant and selects one solution.

20 practice questions 2 video lessons Theory + worked examples

Get started with differential equations in NSW Year 12 Mathematics Extension 1. A differential equation relates a function to its derivatives, and its solution is a function rather than a single number.

You will learn to identify the order of a differential equation from its highest derivative, distinguish first- and second-order equations, and recognise that a general solution can describe a whole family of curves — the foundation for solving them in Extension 1.

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Practice questions

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Watch 2 video(s)

Order and Degree of A Differential Equation (simple and easy explanation) Watch
Differential equation introduction | First order differential equations | Khan Academy Watch

Theory

A differential equation relates an unknown function to its derivatives; its order is the highest derivative present. This NSW Year 12 Mathematics Extension 1 topic (NESA outcome ME1-12-05) introduces solution families and initial value problems.

A differential equation (DE) is an equation involving an unknown function and one or more of its derivatives. Its order is the order of the highest derivative present; its degree is the power of that highest derivative.

A solution is a function that satisfies the DE. A first order DE has infinitely many solutions — a family of curves, one for each value of an arbitrary constant.

An initial condition (a point on the curve) picks out one member of the family — an initial value problem (IVP).

NESA link. Part of the Year 12 Further applications of calculus focus area, outcome ME1-12-05 ("applies calculus to solve problems involving polynomials, further rates of change, areas and volumes and differential equations") with MAO-WM-01. For example, $x^{2} \frac{d y}{d x} = \tan y$ is first order, while $\frac{d^{2} y}{d x^{2}} - 2 \frac{d y}{d x} = e^{x}$ is second order.

The general solution is a family — one curve per constant.

An initial condition selects one particular solution.

order = highest derivative, degree = power of that derivative .

order = highest derivative; degree = its power

The general solution carries an arbitrary constant; an initial condition fixes it. To verify a proposed solution, substitute it and its derivatives and check both sides agree.

Family of solutions. Because integrating introduces a constant, a first order DE has infinitely many solutions until an initial condition is supplied.

How to classify or verify

Order and degree: find the highest derivative (order) and the power it is raised to (degree).
To verify a solution, differentiate it as many times as needed.
Substitute the function and its derivatives into the DE.
Confirm both sides are equal (for all $x$ , or solve for a parameter).

Example 1 — Classify

Classify

\frac{d^{2} y}{d x^{2}} + {(\frac{d y}{d x})}^{3} = x

Solution

The highest derivative is $\frac{d^{2} y}{d x^{2}}$ , to the first power.

second order, first degree

Second order, first degree.

Example 2 — Find a parameter

y = e^{a x}

satisfies

y^{″} - y^{'} - 6 y = 0

. Find

a

Solution

$y^{'} = a e^{a x}$ , $y^{″} = a^{2} e^{a x}$ .

a^{2} - a - 6

=

0 \Rightarrow (a - 3) (a + 2) = 0

a = 3 or a = -2

$a = 3$ or $a = - 2$ .

Example 3 — Infinitely many solutions

Show

y = A e^{- 2 x}

satisfies

\frac{d y}{d x} = - 2 y

for any

A

, and explain why there are infinitely many solutions.

Solution

\frac{d y}{d x}

=

- 2 A e^{- 2 x} = - 2 y

dy/dx = -2y for every A, so infinitely many solutions

Every value of $A$ gives a solution, so there are infinitely many.

Example 4 — A trig solution

y = \sin n x

(

n > 0

) satisfies

y^{″} + 16 y = 0

. Find

n

Solution

$y^{″} = - n^{2} \sin n x$ .

- n^{2} + 16

=

0 \Rightarrow n^{2} = 16

n = 4

$n = 4$ .

Common pitfalls

Confusing order and degree. Order is the highest derivative; degree is the power of that highest derivative.

Forgetting the constant. The general solution has an arbitrary constant — one initial condition fixes it.

Verifying carelessly. Substitute the function and its derivatives, and confirm both sides match.

One solution only. A first order DE has a whole family of solutions, not a single curve.

Frequently asked questions

What is a differential equation?

An equation relating an unknown function to one or more of its derivatives.

What is the order of a differential equation?

The order of the highest derivative that appears (degree is the power of that derivative).

Why are there infinitely many solutions?

Integrating introduces an arbitrary constant, so a first order DE has a family of solutions — one per constant.

What is an initial value problem?

A DE together with an initial condition (a point on the curve) that fixes the constant and selects one solution.

How do you verify a solution?

Differentiate the candidate, substitute it and its derivatives into the DE, and check both sides agree.

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Differential equations – introduction & order

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Common pitfalls

Frequently asked questions

Theory