Differential Equations • Topic 1 of 3

Differential Equations: Order, Degree & Formation

A differential equation relates a function to its derivatives. For example $\dfrac{dy}{dx}=2x$ or $\dfrac{d^2y}{dx^2}+y=0$.

Order and degree

The order is the order of the highest derivative present.
The degree is the power of the highest-order derivative, after the equation is made free of radicals and fractions in the derivatives. Degree is defined only when the equation is polynomial in its derivatives.

For instance, $\left(\dfrac{d^2y}{dx^2}\right)^3+\dfrac{dy}{dx}=0$ has order $2$ and degree $3$.

General and particular solutions

A solution containing arbitrary constants is the general solution; fixing the constants using given conditions gives a particular solution. An $n$th-order equation has $n$ arbitrary constants in its general solution.

Forming a differential equation

Given a family of curves with $n$ arbitrary constants, differentiate $n$ times and eliminate the constants. The result is a differential equation of order $n$ that the whole family satisfies.

Solved Examples

Worked Example

Find the order and degree of $\dfrac{d^2y}{dx^2}+3\left(\dfrac{dy}{dx}\right)^2+y=0$.

Solution

Highest derivative is $\dfrac{d^2y}{dx^2}$ (order $2$); its power is $1$, so degree $1$.

Worked Example

Find the order and degree of $\left(\dfrac{dy}{dx}\right)^3+2y=x$.

Solution

Highest derivative is first order; its highest power is $3$. Order $1$, degree $3$.

Worked Example

Form the differential equation of the family $y=mx$ ($m$ arbitrary).

Solution

Differentiate: $\dfrac{dy}{dx}=m$. Eliminate $m$ using $m=\dfrac{y}{x}$: $\dfrac{dy}{dx}=\dfrac{y}{x}$, i.e. $x\dfrac{dy}{dx}=y$.

Worked Example

Verify that $y=e^{2x}$ is a solution of $\dfrac{dy}{dx}=2y$.

Solution

$\dfrac{dy}{dx}=2e^{2x}=2y$. The equation is satisfied, so $y=e^{2x}$ is a solution.

Key Points

Order = highest derivative present; degree = power of that derivative (polynomial form).
General solution has $n$ arbitrary constants for order $n$; particular solution fixes them.
To form a DE from a family with $n$ constants: differentiate $n$ times, eliminate constants.
Verify a solution by substituting it back into the equation.

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.The degree of $\left(\dfrac{d^2y}{dx^2}\right)^3+\dfrac{dy}{dx}=0$ is:

Explanation: Highest-order derivative (2nd) is raised to power $3$.

Q2.The order of $\dfrac{dy}{dx}+y=x$ is:

Explanation: Highest derivative is first order.

Q3.The DE of the family $y=mx$ is:

Explanation: Eliminate $m$: $dy/dx=y/x$.

Q4.A general solution of an order-$n$ DE has how many arbitrary constants?

Explanation: Exactly $n$ constants.

Practice more MCQs → 📝 Assignment · 35 marks