1.2. Ordinary Differential Equations#
For an \(n\)-th order ordinary differential equation, we can write it in
General Form
(1.2)#\[ F(t, y, y', y'', \ldots, y^{(n)}) = 0 \]
Normal form
(1.3)#\[ \diff[n]{y}{t} = f(t, y', y'', \ldots, y^{(n-1)}) \]
Remark 1.4 (Interval of definition \(I\))
For a problem governed/represented by an ODE, we need to give the actual ODE in form (1.2) or (1.3), besides we also need to give the interval \(I\) that defines the problem. We call this interval \(I\):
the interval of definition,
the interval of existence,
the interval of validity,
or the domain of the solution.
The interval \(I\) can be
an open interval \((a, b)\),
a closed interval \([a, b]\),
an infinite interval \((a, \infty)\),
and so on.
Definition 1.2 (Explicit Solution of an ODE)
A function \(\phi(t)\) that when substituted for \(y\) in equation (1.2) or (1.3) satisfies the equation for all \(t\) in the interval \(I\) is called an explicit solution to the equation on \(I\).
Example 1.4
\(\phi(t)=t^2 - t^{-1}\) is an explicit solution to the equation
\[ \diff[2]{y}{t} - \frac{2y}{t^2}=0 \]\(\phi(t)=c_1 e^{-t} + c_2 e^{2t}\) is an explicit solution to the equation
\[ y'' - y' - 2y=0 \]
Definition 1.3 (Implicit Solution of an ODE)
A relation \(G(t, y) = 0\) is said to be an implicit solution of an ordinary differential equation (1.2) on an interval \(I\), provided that there exists at least one function \(\phi\) that satisfies the relation as well as the differential equation on \(I\).
Example 1.5
The relation \(t^2+y^2=25\) is an implicit solution to the equation
\[ \diff{y}{t} = -\frac{t}{y} \]on the open interval \((-5, 5)\).
The relation \(y^2 - t^3 + 8=0\) implicitly defines a solution to the nonlinear equation
\[ \diff{y}{t}=\frac{3t^2}{2y} \]
Definition 1.4 (Initial Value Problem)
For an \(n\)th order differential equation
if we specify the value of \(y\), \(y'\), …, \(y^{(n-1)}\) at the lower bound \(t_0\) of its interval of definition \(I\) as
where \(y_0\), \(y_1\), … , \(y_{n-1}\) are given constants, then we call this problem an initial value problem (or Cauchy problem). The values specified in (1.5) are called initial conditions.
Definition 1.5 (Lipschitz Continuity)
For a real-value function \(f: I \subseteq \R \rightarrow \R\), if there is a real non-negative constant \(K\) such that
then \(f\) is Lipschitz continuous, and the inequality (1.6) is called the Lipschitz continuity condition. We can also say \(f\) satisfies the Lipschitz condition, or simply call \(f\) a Lipschitz function.
Theorem 1.1 (Existence and Uniqueness)
Consider the initial value problem
If \(f\) is continuous in \(t\) and Lipschitz continuous in \(y\) in some rectangle
that contains the point \((t_0, y_0)\), then the initial value problem has a unique solution \(y=\phi(t)\) in some interval \([t_0 - \delta, t_0 + \delta]\), where \(\delta\) is a positive number.
Remark 1.5
The conditions stated in Theorem 1.1 are sufficient but not necessary. When these conditions are not satisfied, we may see different scenarios for an IVP:
it has no solution,
it has a solution and the solution is unique,
or it has multiple solutions.
In the rest of our discussions, we will assume Theorem 1.1 holds.
1.2.1. Systems of Differential Equations#
A system of ordinary differential equations includes two or more equations involving the derivatives of two or more unknown functions of a single independent variable. For example, if \(x\) and \(y\) denote dependent variables and \(t\) denotes the independent variable, then a system of two first-order differential equations is given by
A solution to a system such as the above one is a pair of differentiable functions \(x=\phi_1(t)\), \(y=\phi_2(t)\), defined on a common interval \(I\), that satisfy each equation of the system on this interval.