Benjamin's Maths World

Definition of ODEs

Ordinary Differential Equations (ODEs) are equations in which the unknown, dependent variable \( y = y(x) \) depends on only one independent variable \( x \), and all derivatives appearing in these equations are “ordinary“: \( y \) can be differentiated any time but against \( x \) only, including \( dy/dx, d^2y/dx^2, d^3y/dx^3, \ldots \) (or denoted by \( y’, y^{\prime\prime}, y^{(3)}\)), and so on. Examples of ODEs are:

1. \( dy/dx + (\sin x)y = e^{-x} \)
2. \( d^2y/dx^2 + 4y = 0 \)
3. \( dy/dx + y^2 = 1 \)

There are three classifications of ODEs: Order, Linearity, and Homogeneity.

Order

Order refers to the highest order of ordinary derivatives appearing in the ODE. For example:

1. \( \color{red}{dy/dx} + 2y = 3\sin^3 x \) is first-order
2. \( 2\color{red}{d^2y/dx^2} + x^5 dy/dx -y^3 = 1 \) is second-order
3. \( x^4\color{red}{d^3y/dx^3} + (d^2y/dx^2)^5 + 7y^5 = e^{-x} \) is third-order

We usually arrange the terms in decreasing order of derivatives.

Linearity

An ODE is called linear if, within each additive term, only at most one out of the dependent variable \( y \) and its ordinary derivatives \( dy/dx, d^2y/dx^2, d^3y/dx^3, \ldots \) is present with a power of \( 1 \). Otherwise, when \( y \) and its derivatives appear as a product together, have powers other than \( 1 \), or are embedded in other non-linear functions, it is called non-linear. For instance,

1. \(x^2d^3y/dx^3 + 5y = -2x \)
2. \( 3dy/dx + (\cos^2 x) y = e^{-x}\tan x \)

are both linear, while

1. \( \color{red}{ydy/dx} = 3x \)
2. \( dy/dx -\color{red}{\cos y} = 2 \)
3. \( \color{red}{(dy/dx)^2} + 4\color{red}{y^5} = 0 \)

are all non-linear.

Homogeneity

A homogeneous ODE (particularly, for the linear one) means that there is no term involving the independent variable \( x \) only (including any non-zero constant as well). Else, it is known as non-homogeneous. For example,

\( dy/dx -xy = 0 \) is homogeneous, but

\( dy/dx = \alpha x^2 + 1 \) with \( \alpha \) being a constant, is always non-homogenous even when \( \alpha = 0 \).

Exercises

You can check that the three ODEs shown in the beginning are

1. First-order, linear, non-homogeneous
2. Second-order, linear, homogeneous
3. First-order, non-linear

Now try to classify the following ODEs according to the three categories: (Beware of how they are arranged!)

1. \( d^2y/dx^2 -2dy/dx -3y =0 \)
2. \( dy/dx + x^4y = 2 \)
3. \( yd^3y/dx^3 + 4x^2y = e^{-x} \)
4. \( dy/dx = x(1-xe^{-y}) \)
5. \( dy/dx + (\cos x) y + e^{-x^2} = 0 \)
6. \( \cot(d^2y/dx^2) = e^{-x} \)

Answers

1. Second-order, linear, homogeneous
2. First-order, linear, non-homogeneous
3. Third-order, non-linear
4. First-order, non-linear
5. First-order, linear, non-homogeneous
6. Second-order, linear, non-homogeneous (just take \( \cot^{-1} \) on both sides)