Chapter 3 Introduction to ordinary differential equations

Differential equations are very important in science and engineering. In this course, we focus on a specific class of differential equations called ordinary differential equations (ODEs). Ordinary refers to dealing with functions of one independent variable. We initially focus on scaler functions of the form \(f: \mathbb{R} \rightarrow \mathbb{R}\). Later in the course, we discuss systems of ODEs and consider vector functions of the form \(f: \mathbb{R} \rightarrow \mathbb{R}^n\). We assume \(f(x)\) is differentiable upto order \(k\). An ODE is an equation for the function \(f(x)\) that involves the function, its derivatives and the independent variable. A general form for an ODE of order \(k\) is \[G(x, f(x), \frac{df}{dx}, ..., \frac{d^kf}{dx^k}) = 0,\] where the highest derivative present in the equation \(G\) is of the order \(k\). The degree of the ODE is the power of highest derivative (when fractional powers have been removed). The ODE is called linear if G is a linear function of \(f(x)\) and its derivatives. This form for the ODE is the so called implicit form. In an explicit form for an ODE, the highest order derivative is given as function of the lower derivatives: \[\frac{d^kf}{dx^k} = F(x, f(x), \frac{df}{dx}, ..., \frac{d^{k-1}f}{dx^{k-1}}) = 0.\]

Example 3.1 Consider the following ODE for the function \(f(x)\): \[ \frac{d^2f}{dx^2} = 5 \left[ 1 + \left( \frac{df}{dx}\right) ^2\right] ^\frac{1}{3} \]

This is a nonlinear explicit ODE of degree 3 and order 2.

Solving an ODE is the task of finding \(f(x)\) such that the ODE is satisfied over the domain of \(x\) (e.g. \(\mathbb{R}\)).

ODEs appear naturally in many areas of sciences and humanities. In the following we provide some examples.

Example 3.2 (Second Newton Law)

Mechanics: A very short introduction

Kinematics is a branch of mechanics that describes the motion of points (objects) without considering the forces that cause them to move. In one dimension \(x(t)\) denotes the position of a particle at time \(t\). Then \(\frac{dx}{dt} = \dot{x} = v\) is defined as the velocity of the particle and \(\frac{d^2x}{dt^2} = \ddot{x} = a\) is defined as acceleration. This can be generalised to higher dimensions using vectors of location, velocity and acceleration.

Dynamics is the branch of mechanics concerned with the study of forces and their effects on motion. Isaac Newton came up with the fundamental physical laws, which govern dynamics in physics:

-First law an object not acted upon by any force either remains at rest or continues to move at a constant velocity

-Second law the vector sum of the forces \(F\) on an object is equal to the mass \(m\) of that object multiplied by its acceleration \(a\): \(F = ma\) \[m \frac{d^2x}{dt^2} = F(t, x, \frac{dx}{dt})\] This is a second order ODE for the position of the object \(x(t)\).

-Third law when one body exerts a force on a second body, the second body exerts a force equal in magnitude and opposite in direction on the first body.

Mechanics used to be thaught until recently in our first year Mathematics course as it provides many links to different areas of mathemaitcs. If you have any doubts, look at this video, for a very cool counting problem for colliding particles and its very unexpected solution and link to mathematics.

Example 3.3 (Population dynamics: Malthus (1798))
Consider \(P(t)\) denotes the population of certain species at time \(t\). Malthus proposed the following simple ODE: \[\frac{dP}{dt} = kP,\] with \(k>0\), this ODE results in an exponential increase in the population in time.
Example 3.4 (Population dynamics: Logistic Growth (Verhulst, 1845))
Verhgulst proposed a modification to Malthus law, creating a carrying capacity \(C\) for the population: \[\frac{dP}{dt} = kP\left(1- \frac{P}{C}\right).\]
Example 3.5 (Radius of curvature)

In geometry, given radius of curvature \(R(x, y)\), we can find equation for the curve \(y(x)\) using the following ODE that definition for the radius of curvature. \[R(x, y) = \frac{\left(1 + (\frac{dy}{dx})^2\right)^{3/2}}{\frac{d^2y}{dx^2}}.\]

3.1 Particular and General Solutions

\(f_{PI}(x)\) is called a Particular Integral or Particular Solution of an ODE such that \[G(x, f(x), \frac{df}{dx}, ..., \frac{d^kf}{dx^k}) = 0,\] is satisfied over the domain \(x \in \mathbb{R}\).

\(f_{GS}\) is called a General Solution of an ODE of the order \(k\), if \[f_{GS} = f_{GS}(x; c_1, c_2, \cdots, c_k)\] is a general family of solutions that fulfil the ODE. The parameters \(\{c_i\}^k_{i=1}\) are the constants of integration and are usually fixed by initial or boundary conditions. In this course we, concern ourselves with methods that allow us to obtain such solutions. Rigourous mathematical results on existance and uniqueness of such solutions are discussed in the second year.

Example 3.6 (from kinematics) Object moving with constant speed \(v\): \[\frac{dx}{dt} = v.\]

One particular solution is \(x_{PI} = vt\) and another one is \(x_{PI} = vt + 1\). The general solution is \(x_{GS} = vt + c_1\), where \(c_1\) is the constant of integration.

If we are also told that \(x(t=0)=x_0\), we have \(x(t) = vt + x_0\), which is the solution to the initial value problem.