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Differential Equations
Introduction
Basic Concepts
Practice: Order and Degree
General and Particular Solutions of a Differential Equation
Practice: Verifying Solutions
Methods of Solving First Order, First Degree Differential Equations
Practice: Variables Separable
Practice: Homogeneous Equations
Practice: Linear Differential Equations
Miscellaneous Examples
Chapter Review: Mixed Practice
Summary
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Glossary

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Differential EquationsBasic Concepts

Reading time: ~20 min

You have solved plenty of equations already, for example

x^2 - 3x + 3 = 0 \qquad \ldots (1)

\sin x + \cos x = 0 \qquad \ldots (2)

x + y = 7 \qquad \ldots (3)

In each of them the unknowns are numbers. Now look at

x\dfrac{dy}{dx} + y = 0 \qquad \ldots (4)

Here y is a function of x, and the equation contains y together with its derivative \frac{dy}{dx}. We call x the independent variable and y the dependent variable.

An equation involving the derivative (or derivatives) of the dependent variable with respect to the independent variable (or variables) is called a differential equation.

So y'' + 4y = \sin x is , because it contains a derivative, while x^2 + y^2 = 4 is : no derivative appears in it.

When the dependent variable depends on only one independent variable, as in (4), the equation is called an ordinary differential equation. For example,

2\dfrac{d^2y}{dx^2} + \left(\dfrac{dy}{dx}\right)^3 = 0 \qquad \ldots (5)

is ordinary. Equations with derivatives with respect to more than one independent variable are called partial differential equations; they are beyond this course, so in this chapter "differential equation" always means an ordinary one.

Writing \frac{dy}{dx} again and again gets heavy, so we use short names for the derivatives:

\dfrac{dy}{dx} = y', \quad \dfrac{d^2y}{dx^2} = y'', \quad \dfrac{d^3y}{dx^3} = y'''

Beyond the third derivative so many dashes would be confusing, so the nth derivative \frac{d^ny}{dx^n} is written y_n.

Order of a differential equation

The first thing to read off any differential equation is the highest derivative it contains.

The order of a differential equation is the order of the highest order derivative of the dependent variable (with respect to the independent variable) that occurs in it.

\dfrac{dy}{dx} = e^x \qquad \ldots (6)

\dfrac{d^2y}{dx^2} + y = 0 \qquad \ldots (7)

\left(\dfrac{d^3y}{dx^3}\right) + x^2\left(\dfrac{d^2y}{dx^2}\right)^3 = 0 \qquad \ldots (8)

Equation (6) contains only the first derivative, so its order is . The highest derivative in (7) is \frac{d^2y}{dx^2}, so its order is . In (8) the cube sits on \frac{d^2y}{dx^2}, but the highest derivative present is \frac{d^3y}{dx^3}, so the order of (8) is .

Powers do not change the order: only which derivative is the highest matters.

Degree of a differential equation

The degree only makes sense under one condition: the equation must be a polynomial equation in its derivatives y', y'', y''' and so on. Compare these three:

\dfrac{d^3y}{dx^3} + 2\left(\dfrac{d^2y}{dx^2}\right)^2 - \dfrac{dy}{dx} + y = 0 \qquad \ldots (9)

\left(\dfrac{dy}{dx}\right)^2 + \dfrac{dy}{dx} - \sin^2 y = 0 \qquad \ldots (10)

\dfrac{dy}{dx} + \sin\left(\dfrac{dy}{dx}\right) = 0 \qquad \ldots (11)

Equation (9) is a polynomial in y''', y'' and y'. Equation (10) is a polynomial in y'; the \sin^2 y does no harm, because the condition is only about the derivatives, not about y itself. Equation (11) a polynomial in y', because \sin y' cannot be written as a sum of powers of y'.

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

So equations (6), (7), (8) and (9) each have degree , equation (10) has degree because its highest derivative y' appears squared, and the degree of (11) is not defined.

Note

The order and the degree (when the degree is defined) of a differential equation are always positive whole numbers. Read the order first; the degree is the power of that derivative. A bigger power sitting on a lower derivative does not count.

Read the Equation

Tap the chips to cycle through the choices and set the order and the degree of each equation. A row turns green once both are right.

Example 1 : Find the order and degree, if defined, of each of the following differential equations:

(i) \dfrac{dy}{dx} - \cos x = 0

(ii) xy\dfrac{d^2y}{dx^2} + x\left(\dfrac{dy}{dx}\right)^2 - y\dfrac{dy}{dx} = 0

(iii) y''' + y^2 + e^{y'} = 0

Solution :

(i) The highest derivative is \frac{dy}{dx}, so the order is . The equation is a polynomial in y', and \frac{dy}{dx} appears to the power 1, so the degree is .

(ii) The highest derivative is \frac{d^2y}{dx^2}, so the order is . The equation is a polynomial in y'' and y', and \frac{d^2y}{dx^2} appears to the power 1, so the degree is . The square belongs to \frac{dy}{dx}, a lower derivative.

(iii) The highest derivative is y''', so the order is . But the term e^{y'} is not a polynomial in y', so the degree is .

Speed Round: Order and Degree

Order first, then the power of that derivative. Watch out for a sine, a log or an exponential of a derivative, and for square roots that have to be squared away first.

Get five right to move on.

Sina