Ordinary differential equations

Ordinary differential equations (ODEs) represent one of the most powerful and widely applicable areas of mathematics, providing the fundamental language for describing dynamic systems across science and engineering. Within the field of differential equations, linear constant coefficient differential equations (LCCDEs) in particular occupy a special position due to their mathematical tractability and widespread practical importance. These equations, while representing a specific subset of all possible differential equations, appear with remarkable frequency in real-world applications and serve as the foundation for understanding more complex systems.

Definition and classification of differential equations

A differential equation is a mathematical equation that relates a function to its derivatives. In the context of ordinary differential equations, we deal with functions of a single independent variable (typically time $t$ or position $x$) and their derivatives.

The general form of an $n$-th order ODE can be written as:

$$F\left(x,y,y^{\prime },y^{\prime \prime },\dots ,y^{(n)}\right)=0$$

where $y=y(x)$ is the unknown function, and $y^{\prime },y^{\prime \prime },\dots ,y^{(n)}$ represent its first, second, through $n$-th derivatives.

Definitions

Order

The highest derivative that appears in the equation.

Degree

The highest power of the highest-order derivative (when the equation is polynomial in derivatives).

Linear vs. nonlinear equations

A differential equation is linear if it can be written in the form:

$$a_n(x)y^{(n)}+a_{n-1}(x)y^{(n-1)}+\cdots +a_1(x)y^{\prime }+a_0(x)y=f(x)$$

Key characteristics of linear equations include:

• The dependent variable $y$ and all its derivatives appear to the first power only.
• No products of $y$ and its derivatives appear.
• No transcendental functions of $y$ or its derivatives appear.

Linear constant coefficient differential equations (LCCDEs)

A linear constant coefficient differential equation (LCCDE) is a linear ODE where all coefficients are constants (independent of the variable). The general form of an $n$-th order LCCDE is:

$$a_n{y}^{(n)}+a_{n-1}{y}^{(n-1)}+\cdots +a_1{y}^{\prime }+a_{0}y=f(x)$$

where $a_n,a_{n-1},\dots ,a_1,a_0$ are constants with $a_n\ne 0$.

Definitions

Homogeneous LCCDE

An LCCDE where $f(x)=0$: $a_n{y}^{(n)}+a_{n-1}{y}^{(n-1)}+\cdots +a_1{y}^{\prime }+a_{0}y=0$.

Non-homogeneous LCCDE

An LCCDE where $f(x)\ne 0$: $a_n{y}^{(n)}+a_{n-1}{y}^{(n-1)}+\cdots +a_1{y}^{\prime }+a_{0}y=f(x)$.

First-order linear constant coefficient equations

Homogeneous case

The simplest LCCDE is the first-order homogeneous equation:

$$\frac{dy}{dx}+ay=0$$

or equivalently:

$$y^{\prime }+ay=0$$

The solution is:

$$y=C{e}^{-ax}$$

where $C$ is an arbitrary constant determined by initial conditions.

Non-homogeneous case

The first-order non-homogeneous LCCDE has the form:

$$y^{\prime }+ay=f(x),$$

which has the following solution:

$$y=e^{-ax}\left[\int e^{ax}f(x)dx+C\right]$$

Second-order linear constant coefficient equations

Second-order LCCDEs are particularly important because they model many fundamental physical phenomena, including oscillatory motion, electrical circuits, and vibrating systems.

General second-order LCCDE

$$a{y}^{\prime \prime }+b{y}^{\prime }+cy=f(x)$$

where $a$, $b$, and $c$ are constants with $a\ne 0$.

The complete solution consists of two parts:

$$y=y_h+y_p$$

where:

• $y_h$ is the homogeneous solution (solution to $a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$)
• $y_p$ is a particular solution to the non-homogeneous equation

The characteristic equation method provides a systematic approach to finding the homogeneous solution of LCCDEs.

Derivation of the characteristic equation

For the homogeneous equation $a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$, we assume a solution of the form:

$$y=e^{rx}$$

where $r$ is a constant to be determined.

Computing the derivatives:

• $y^{\prime }=r{e}^{rx}$
• $y^{\prime \prime }=r^2{e}^{rx}$

Substituting into the differential equation:

$$a(r^2{e}^{rx})+b(r{e}^{rx})+c(e^{rx})=0$$

Factoring out $e^{rx}$:

$$e^{rx}(a{r}^2+br+c)=0$$

Since $e^{rx}\ne 0$ for any finite $r$ and $x$, we must have:

Characteristic equation

$$a{r}^2+br+c=0$$

Also called the auxiliary equation.

Solving the characteristic equation

The characteristic equation is a quadratic equation with solutions:

$$r=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

The discriminant $\Delta =b^2-4ac$ determines the nature of the solutions and the behavior of the system.

The behavior of second-order systems is classified based on the discriminant of the characteristic equation. This classification has profound physical significance and appears across many disciplines.

Case 1: overdamping ($\Delta >0$, $b^2>4ac$)

Characteristic roots: two distinct real roots $r_1$ and $r_2$

$$r_1=\frac{-b+\sqrt{b^2-4ac}}{2a},r_2=\frac{-b-\sqrt{b^2-4ac}}{2a}$$

General solution:

$$y_h=C_1{e}^{r_{1}x}+C_2{e}^{r_{2}x}$$

Physical characteristics:

• No oscillations: the solution approaches equilibrium monotonically.
• Slow return to equilibrium: high damping prevents oscillations but slows response.
• Two time constants: the system has two characteristic time scales.

Case 2: critical damping ($\Delta =0$, $b^2=4ac$)

Characteristic roots: one repeated real root $r=-\frac{b}{2a}$

General solution:

$$\begin{array}{rl}{y}_h& =(C_1+C_{2}x)e^{rx}\\ & =(C_1+C_{2}x)e^{-\frac{b}{2a}x}\end{array}$$

Physical characteristics:

• Fastest return to equilibrium without oscillation.
• Optimal damping: represents the boundary between oscillatory and non-oscillatory behavior.
• Engineering significance: often the desired response in control systems.

Case 3: underdamping ($\Delta <0$, $b^2<4ac$)

Characteristic roots: complex conjugate pair $r=\alpha \pm \beta i$

$$\alpha =-\frac{b}{2a},\beta =\frac{\sqrt{4ac-b^2}}{2a}$$

General solution:

$$y_h=e^{\alpha x}(C_1\cos (\beta x)+C_2\sin (\beta x))$$

Physical characteristics:

• Oscillatory motion: the solution exhibits sinusoidal behavior.
• Exponential envelope: the amplitude of oscillations changes exponentially.
• Two frequencies: natural frequency and damped frequency.

The solution can also be written as:

$$y_h=A{e}^{\alpha x}\cos (\beta x-\varphi )$$

where $A=\sqrt{C_1^2+C_2^2}$ and $\varphi =\arctan (C_2/C_1)$.

Physical interpretation:

• $\alpha$ determines the decay rate of the oscillations.
• $\beta$ determines the frequency of oscillation.
• If $\alpha <0$, oscillations decay (stable system).
• If $\alpha >0$, oscillations grow (unstable system).
• If $\alpha =0$, oscillations maintain constant amplitude.

Laplace transforms

Traditional methods of solving nonhomogeneous differential equations by hand are usually difficult and/or time consuming. Laplace transforms can be used to reduce many solution procedures to simple algebra.

Laplace transform

$$F(s)={\int }_0^{\infty }f(t)e^{-st}dt$$

Every mathematical function, $f(t)$, has a Laplace transform, written as $F(s)$ or $\mathcal{L}(s)$. The transform is written in the $s$-domain, regardless of the independent variable in the original function. The variable $s$ is equivalent to a derivative operator, although it may be handled in the equations as a simple variable. This equation converts a function into its Laplace transform.

Sidenote

Using transform tables

It is generally unnecessary to actually compute a function’s Laplace transform. Tables of transform pairs are readily available for common functions — see the reference table below.

The following table lists common Laplace transform pairs, used as a quick reference when transforming functions to and from the $s$-domain.

$f(t)$	$F(s)$
$\delta (t)$, impulse at $t=0$	1
$u(t)$, step at $t=0$	$1/s$
$t[u(t)]$, ramp at $t=0$	$1/s^2$
$e^{-\alpha t}$	$1/(s+\alpha )$
$t{e}^{-\alpha t}$	$1/(s+\alpha {)}^2$
$e^{-\alpha t}\sin \beta t$	$\beta /[(s+\alpha {)}^2+{\beta }^2]$
$e^{-\alpha t}\cos \beta t$	$(s+\alpha )/[(s+\alpha {)}^2+{\beta }^2]$
$\frac{d^{n}f(t)}{d{t}^n}$	$s^{n}F(s)-{\sum }_{m=0}^{n-1}{s}^{n-m-1}\frac{d^{m}f(0)}{d{t}^m}$
${\int }_0^{t}f(\tau )d\tau$	$(1/s)F(s)$
${\int }_0^{t}x(t-\tau )h(t)d\tau$	$H(s)X(s)$
$f(t-\tau )u(t-\tau )$	$e^{-\tau s}F(s)$
:::

Inverse Laplace transform

$$f(t)=\frac{1}{2\pi j}{\int }_{\sigma -j\infty }^{\sigma +j\infty }F(s)e^{st}dt$$

Extracting a function from its transform is the inverse Laplace transform operation. Although the equation above could be used and other methods exist, this operation is almost always done using a table.

Initial value theorem (IVT):

$$\underset{s\to \infty }{\lim }sF(s)$$

Final value theorem (FVT):

$$\underset{s\to 0}{\lim }sF(s)$$