Last modified: September 13, 2022
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An ordinary differential equation (ODE) is an equation that involves:
I. One independent variable, often denoted by $t$ (in many applications, $t$ represents time).
II. One dependent variable (or unknown function), which we may denote by $y(t)$.
III. The derivatives of the dependent variable with respect to the independent variable.
Formally, an ODE can be written as
$$F\bigl(t, y(t), y'(t), y''(t), \dots, y^{(n)}(t)\bigr) = 0$$
where $y^{(n)}(t)$ denotes the $n$-th derivative of $y$ with respect to $t$. The integer $n$ is called the order of the ODE.
The term ordinary differentiates it from a partial differential equation (PDE), in which derivatives with respect to multiple independent variables can appear.
A (single) ODE is linear if it can be expressed in the form
$$a_n(t)y^{(n)}(t) + a_{n-1}(t)y^{(n-1)}(t) + \cdots + a_1(t)y'(t) + a_0(t)y(t) = g(t)$$
where $a_0(t), \dots, a_n(t)$ and $g(t)$ are functions of $t$ only (i.e., do not depend on $y$ or its derivatives). If any product of the dependent variable and/or its derivatives or any power other than $1$ of $y$ or its derivatives appears, then the ODE is nonlinear.
Differential equations capture the relationship between a function (representing a quantity of interest) and its rates of change. These arise naturally in numerous domains:
To find a unique solution, one often needs:
The combination of a differential equation and enough conditions to fix a unique solution is called an initial value problem (IVP) or boundary value problem (BVP), depending on whether the conditions are specified at a single point (IVP) or at different points (BVP).
A crucial theoretical aspect of ODEs is ensuring whether a solution to a given IVP exists and whether it is unique. One fundamental result for first-order ODEs is the Picard–Lindelöf theorem (also known as the Existence and Uniqueness Theorem), which states that if:
I. $f(t,y)$ is continuous in a region around $(t_0, y_0)$,
II. $f$ satisfies a Lipschitz condition in $y$ (i.e., there exists a constant $L$ such that
$$\bigl| f(t, y_1) - f(t, y_2) \bigr| \le L \bigl| y_1 - y_2 \bigr|$$
for all $y_1, y_2$ in that region), then there exists a time interval $(t_0 - \delta, t_0 + \delta)$ on which there is a unique solution $y(t)$ satisfying $y(t_0) = y_0$.
We revisit the key concepts with further mathematical detail:
If an ODE contains derivatives up to the $n$-th derivative, it is called an $n$-th order ODE. For example:
The degree is determined by writing the ODE in polynomial form in its highest-order derivative. For example, the ODE
$$\left(y''\right)^2 + \left(y'\right)^3 + y = 0$$ is not in polynomial form due to the squared second derivative and cubed first derivative. However, if we could (somehow) algebraically solve for the highest-order derivative and rewrite it linearly or as a polynomial expression without fractional exponents, then the exponent of that highest-order derivative would tell us the degree. Many ODEs (especially linear ones) are understood in simpler terms: the degree is typically $1$ for linear ODEs.
A linear ODE of order $n$ has the form
$$a_n(t)y^{(n)} + a_{n-1}(t)y^{(n-1)} + \cdots + a_1(t)y' + a_0(t)y = g(t),$$
where each $a_k(t)$ (for $k = 0,1,\dots,n$) and $g(t)$ depend only on $t$. No products like $(y')^2$ or $yy''$ occur, nor do terms such as $\sin(y)$. If such terms do occur, the ODE is nonlinear.
$$a_n(t)y^{(n)} + a_{n-1}(t)y^{(n-1)} + \cdots + a_1(t)y' + a_0(t)y = 0.$$ - It is nonhomogeneous (or inhomogeneous) if $g(t) \neq 0$.
Below are some standard forms that frequently appear.
$$\frac{dy}{dt} = f\bigl(t, y(t)\bigr), \quad y(t_0) = y_0$$
$$\frac{dy}{dt} + p(t)y = g(t).$$
This is a subset of the above form but is special because its solution technique is well-known. A classic approach is the Integrating Factor method:
I. Multiply both sides by the integrating factor
$$\mu(t) = e^{\int p(t)dt}.$$
II. Rewrite the left-hand side as the derivative of $\mu(t)y(t)$.
III. Integrate both sides w.r.t. $t$ to solve for $y(t)$.
$$\frac{d^2y}{dt^2} + a(t)\frac{dy}{dt} + b(t)y = g(t)$$
When $a$ and $b$ are constants, the ODE
$$y'' + ay' + by = g(t)$$
can be solved using:
I. Characteristic equation for the associated homogeneous part:
$$r^2 + ar + b = 0$$
II. The solution of the homogeneous ODE depends on the discriminant $\Delta = a^2 - 4b$:
III. A particular solution $y_p(t)$ must be found (e.g., via the method of undetermined coefficients or variation of parameters) for the nonhomogeneous case.
IV. The general solution is $y(t) = y_h(t) + y_p(t)$.
$$\frac{dy}{dt} = f\bigl(y(t)\bigr).$$
Analyzing equilibrium (steady-state) solutions where $f(y)=0$ is a powerful tool for studying the long-term behavior (qualitative analysis).
A solution to an ODE on an interval $I$ is a function $y(t)$ that:
I. Is differentiable up to the required order on $I$.
II. Substitutes into the ODE to satisfy it identically for all $t \in I$.
I. First-Order Linear with Constant Coefficient
$$\frac{dy}{dt} = ay \quad \longrightarrow \quad \frac{dy}{y} = adt$$
Integrating both sides:
$$\ln|y| = at + C \quad \longrightarrow \quad y(t) = C_1 e^{a t}$$
If $y(t_0) = y_0$, then $C_1 = y_0 e^{-a t_0}$.
II. Second-Order Homogeneous with Constant Coefficients
$$y'' + ay' + by = 0.$$ The characteristic equation is $r^2 + ar + b = 0$. Let $\Delta = a^2 - 4b$. - If $\Delta > 0$ with roots $r_1, r_2$, the general solution is
$$y(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t}.$$ - If $\Delta = 0$ with repeated root $r$, the general solution is
$$y(t) = \bigl(C_1 + C_2t\bigr) e^{r t}.$$ - If $\Delta < 0$ with complex roots $\alpha \pm i\beta$, the general solution is
$$y(t) = e^{\alpha t}\bigl(C_1 \cos(\beta t) + C_2 \sin(\beta t)\bigr).$$
III. Second-Order Nonhomogeneous with Constant Coefficients
$$y'' + ay' + by = g(t).$$
One finds the general solution as
$$y(t) = y_h(t) + y_p(t),$$ where $y_h(t)$ is the general solution of the homogeneous equation, and $y_p(t)$ is any particular solution of the original nonhomogeneous equation.
I. Newton’s Second Law of Motion
Often written as $F = ma$. Since $a = \frac{d^2 x}{dt^2}$,
$$m\frac{d^2x}{dt^2} = F(x, t).$$
This is a second-order ODE. If $F$ depends only on $x$ and $t$ (and possibly $v = x'$), it may be nonlinear or linear (if $F$ is linear in $x$, $x'$, etc.).
II. Population Dynamics
The logistic model:
$$\frac{dP}{dt} = rP \left(1 - \frac{P}{K}\right),$$ is a first-order nonlinear autonomous ODE describing population growth with a carrying capacity $K$.
III. RC Circuit (First-Order Linear ODE)
Consider a resistor $R$ in series with a capacitor $C$. The voltage $V_C(t)$ across the capacitor satisfies:
$$C\frac{dV_C}{dt} + \frac{V_C(t)}{R} = \frac{V_{\text{in}}(t)}{R}.$$
Rearranged:
$$\frac{dV_C}{dt} + \frac{1}{RC}V_C(t) = \frac{V_{\text{in}}(t)}{RC}.$$
This is a first-order linear ODE.
ODEs are ubiquitous in mathematical modeling across disciplines: