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 \( dy/dx, d^2y/dx^2, d^3y/dx^3, \ldots \), 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 examples:

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.

Linearity

An ODE is called linear if over each additive term, only 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 the derivatives appear as a product together, have powers other than \( 1 \), or are embedded in other functions, it is called non-linear. For instances,

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 examples,

\( dy/dx -xy = 0 \) 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)