1.1 Differential equations and models
Definition 1.1.1. An ordinary differential equation (ODE) of order \(n\) for a function \(y\) of one variable \(t\) is a relation \[ F\bigl (t, y, y', \dots , y^{(n)}\bigr ) = 0 . \] It is in normal form if it is solved for the highest derivative, \(y^{(n)} = f\bigl (t, y, \dots , y^{(n-1)}\bigr )\). A solution on an interval \(I\) is an \(n\) times differentiable function \(\phi \) on \(I\) such that \[ F\bigl (t, \phi (t), \phi '(t), \dots , \phi ^{(n)}(t)\bigr ) = 0 \qquad \text {for every } t \in I . \] The equation is linear if \(F\) is affine in \(y, y', \dots , y^{(n)}\), and autonomous if \(t\) does not appear explicitly. A system of \(m\) equations for a vector \(y(t) \in \mathbb R^m\) is defined in the same way.
An equation with partial derivatives of a function of several variables is a partial differential equation, the subject of the companion course. The definition insists that a solution is defined on an interval: a solution is a function together with its domain, and we shall see that the domain cannot always be chosen freely.
- (a)
- Malthus. A population growing at a constant per-capita rate \(r\) satisfies \(y' = ry\), with solutions \(y(t) = y_0e^{rt}\).
- (b)
- Verhulst. If crowding reduces the per-capita rate linearly, \(y' = ry(1 - y/K)\), the logistic equation, where \(K\) is the carrying capacity.
- (c)
- Newton’s law of cooling. A body at temperature \(T\) in surroundings at \(T_a\) satisfies \(T' = -k(T - T_a)\).
- (d)
- The pendulum. A pendulum of length \(\ell \) satisfies \(\theta '' + (g/\ell )\sin \theta = 0\), a second-order non-linear equation.
- (e)
- Predator and prey. Lotka and Volterra modelled prey \(x\) and predators \(y\) by \(x' = ax - bxy\), \(y' = -cy + dxy\).
- (f)
- An epidemic. Kermack and McKendrick’s SIR model is \(S' = -\beta SI\), \(I' = \beta SI - \gamma I\), \(R' = \gamma I\).
Theorem 1.1.3. The equation \(y^{(n)} = f\bigl (t, y, y', \dots , y^{(n-1)}\bigr )\) is equivalent to the first-order system \[ x_1' = x_2, \quad x_2' = x_3, \quad \dots , \quad x_{n-1}' = x_n, \quad x_n' = f(t, x_1, \dots , x_n), \] in the sense that \(y\) solves the equation if and only if \(x = (y, y', \dots , y^{(n-1)})\) solves the system.
Proof. If \(y\) solves the equation, then \(x_k = y^{(k-1)}\) satisfies \(x_k' = y^{(k)} = x_{k+1}\) for \(k < n\) and \(x_n' = y^{(n)} = f(t, x)\). Conversely, if \(x\) solves the system, put \(y = x_1\). The first \(n - 1\) equations give \(x_k = y^{(k-1)}\) by induction, so \(y\) is \(n\) times differentiable and \(y^{(n)} = x_n' = f(t, y, \dots , y^{(n-1)})\). □
The theorem is why the general theory can be developed for first-order systems \(x' = f(t, x)\) alone, as it will be from the next section on. The pendulum becomes \(x_1' = x_2\), \(x_2' = -(g/\ell )\sin x_1\), with \(x_1\) the angle and \(x_2\) the angular velocity.
Definition 1.1.4. An initial value problem (IVP) for the system \(x' = f(t, x)\) asks for a solution on an interval containing \(t_0\) with \(x(t_0) = x_0\) prescribed.
For an \(n\)th-order equation the initial data are \(y(t_0), y'(t_0), \dots , y^{(n-1)}(t_0)\), which by Theorem 1.1.3 is the initial state of the equivalent system. A pendulum needs both its initial angle and its initial angular velocity.
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