Solving differential equations with examples

From: 3blue1brown

Differential equations are fundamental to describing the laws of physics and are widely applicable beyond just physical phenomena [00:00:06]. Understanding them provides a new perspective on the world [00:00:17]. This article provides an overview of differential equations, offering a big-picture view while also delving into specific examples [00:00:25]. Basic calculus knowledge (derivatives, integrals) is assumed, with some linear algebra needed for later concepts [00:00:35].

Why Differential Equations Arise [00:00:44]

Differential equations are used when it’s simpler to describe how something changes rather than its absolute state [00:00:44]. For instance, it’s easier to explain why population sizes grow or shrink than their exact values at a specific time [00:00:49]. In Newtonian mechanics, motion is described by force, and force determines acceleration, which is a statement about change [00:01:05].

Types of Differential Equations [00:01:15]

Differential equations generally come in two forms:

• Ordinary Differential Equations (ODEs): Involve functions with a single input, often representing time [00:01:15]. These focus on a finite collection of values changing over time [00:01:46].
• Partial Differential Equations (PDEs): Deal with functions having multiple inputs [00:01:24]. They typically involve a continuum of values changing, such as temperature at every point of a solid body or fluid velocity in space [00:01:35]. Partial differential equations will be explored more closely in subsequent discussions [00:01:30].

Solving a Simple ODE: Projectile Motion [00:02:04]

Physics offers a good starting point for simple examples [00:02:04]. Consider the trajectory of an object thrown in the air [00:02:13]. Near Earth’s surface, gravity causes a downward acceleration of 9.8 meters per second per second (denoted as ‘g’) [00:02:17]. This means that velocity vectors gain an additional downward component of 9.8 m/s every second [00:02:32].

Setting up the Equation [00:02:47]

Focus on the y-coordinate as a function of time. Its first derivative (y-dot) is the vertical velocity, and its second derivative (y-double-dot) is the vertical acceleration [00:02:52]. The equation is: $\frac{d^{2}y}{d{t}^2}=y\text{-double-dot}=-g$ [00:03:15]

Solving by Integration [00:03:22]

To solve this, we integrate backwards:

1. Find Velocity: What function has -g as a derivative? $y\text{-dot}=-gt+v_0$ where $v_0$ is the initial velocity. An extra degree of freedom (the constant of integration) is determined by an initial condition [00:03:32].
2. Find Position: What function has -gt + v_0 as a derivative? $y=-\frac{1}{2}g{t}^2+v_{0}t+y_0$ where $y_0$ is the initial position. Again, an additional constant is determined by the initial position [00:03:48].

This demonstrates how a differential equation is solved by finding a function based on information about its rate of change [00:04:06].

More Complex ODEs: Gravitational Force & Pendulum [00:04:14]

Problems become more complex when forces depend on the body’s position [00:04:14]. For instance, in planetary motion, gravity is not constant but inversely proportional to the square of the distance between bodies [00:04:20]. Here, acceleration is a function of position, creating a “dance between two mutually interacting variables” [00:04:37]. Often, puzzles in differential equations involve finding a function whose derivatives are defined in terms of the function itself [00:04:58].

In physics, second-order differential equations (where the highest derivative is the second derivative) are common [00:05:10]. Solving them feels like an “infinite continuous jigsaw puzzle” [00:05:28].

The Pendulum Example [00:05:53]

The pendulum is a classic example. While often approximated as simple harmonic motion (a sine wave) with a period of $2\pi \sqrt{l/g}$ for small angles, actual pendulums behave differently at larger angles [00:06:04]. The period is longer, and the motion doesn’t look like a sine wave [00:06:29].

Setting up the Pendulum Equation [00:06:54]

Let $\theta$ be the angle with the vertical, and $x=l\theta$ be the distance along the arc [00:06:59]. The component of gravity in the direction of motion (opposite to displacement) is $-g\sin (\theta )$ [00:07:13]. The acceleration along the arc is $x\text{-double-dot}=-g\sin (\theta )$ [00:08:00]. Since $x=l\theta$, we have $x\text{-double-dot}=l\theta \text{-double-dot}$. Thus, the equation for $\theta$ is: $\theta \text{-double-dot}=-\frac{g}{l}\sin (\theta )$ [00:08:27] Adding air resistance (proportional to velocity), represented by a constant $\mu$: $\theta \text{-double-dot}=-\frac{g}{l}\sin (\theta )-\mu \theta \text{-dot}$ [00:08:36] This equation is “particularly juicy” because it’s not easy to solve analytically [00:08:55]. The $\sin (\theta )$ term is precisely why real pendulums don’t oscillate with a sine wave pattern [00:09:16].

Difficulty of Analytic Solutions [00:09:48]

Differential equations are “really freaking hard to solve” [00:09:48]. For the pendulum without damping, an analytic solution exists but is “hilariously complicated,” involving functions rarely seen and complex integrals [00:09:50]. With the damping term, often no known exact analytic solution exists [00:10:23]. This means we often “short circuit” the actual solution process and go directly to building understanding and making computations from the equations [00:10:45].

Understanding without Analytic Solutions: Phase Space [00:11:02]

A powerful way to understand the behavior of systems governed by differential equations is through visualizing their “state” in a phase space [00:11:25].

What is Phase Space? [00:11:37]

The “state” of a pendulum can be described by two numbers: its angle ($\theta$) and its angular velocity ($\theta \text{-dot}$) [00:11:37]. These two values define a point in a two-dimensional phase space [00:11:52]. Each point in this space represents a possible initial condition [00:11:59].

For a damped pendulum, a typical trajectory in phase space is an inward spiral, signifying that peak velocity and displacement decrease with each swing [00:12:18]. The point spirals towards the origin ($\theta =0,\theta \text{-dot}=0$), where the pendulum is still [00:13:04]. It’s crucial to remember that phase space is an abstract representation, distinct from the physical space of the pendulum [00:12:54].

Vector Fields in Phase Space [00:13:25]

A differential equation can be visualized as a vector field in phase space [00:13:25]. The pendulum’s state is a vector $(\theta ,\theta \text{-dot})$ [00:13:31]. Its rate of change (how it moves in this diagram) is given by its derivative: $(\theta \text{-dot},\theta \text{-double-dot})$ [00:13:43]. Since $\theta \text{-double-dot}$ is expressed in terms of $\theta$ and $\theta \text{-dot}$ by the differential equation, every point in phase space has a specific velocity vector [00:14:24].

This effectively breaks a single second-order differential equation into a system of two first-order equations [00:14:56]. Solving the equation means finding a trajectory in this space where the point’s velocity at every moment matches the vector from the field [00:15:28].

Insights from Phase Space [00:16:09]

• High Initial Velocity: Regions where $\theta \text{-dot}$ is high show vectors guiding the point to travel far right before spiraling inward [00:16:09]. This corresponds to a pendulum with enough initial velocity to rotate fully multiple times before settling into decaying oscillations [00:16:19].
• Parameter Effects: Changing the air resistance term ($\mu$) in the equation immediately shows how trajectories spiral inward faster, meaning the pendulum slows down quicker [00:16:33]. This provides intuition that might not be obvious from the equations alone [00:16:51].

Higher Dimensions and Generalization [00:17:19]

Any system of ordinary differential equations can be described by a vector field like this, making it a very general way to understand their behavior [00:17:09]. While the pendulum is 2D, systems like the three-body problem (three masses in 3D space with gravity) have 18 degrees of freedom, resulting in an 18-dimensional phase space [00:17:22].

The concept of phase space has been called “one of the most powerful inventions of modern science” [00:19:23]. It allows for questions about a whole spectrum of initial states, not just a single one [00:19:31]. The collection of all possible trajectories is called phase flow [00:19:40].

Stability Analysis [00:19:50]

Phase space is useful for analyzing stability. Fixed points in the space (where $\theta$ and $\theta \text{-dot}$ are zero, representing a still pendulum, either hanging down or perfectly balanced upright) can be assessed for stability: do tiny nudges cause the system to return to the fixed point or move away from it? [00:19:52] This is intuited by observing whether the phase flow tends to contract or expand around the fixed point [00:20:28].

Applications Beyond Physics: Love Dynamics [00:20:44]

Differential equations extend beyond physics [00:20:56]. Stephen Strogatz’s work on modeling affection provides a whimsical example [00:20:50].

Imagine a scenario where:

• Your affection increases when your companion shows interest and decreases when they seem colder [00:21:19].
• Your companion is attracted when you seem uninterested but turned off when you are too keen [00:21:35].

The phase space for these “romance equations” can look very similar to the pendulum diagram, showing endless cycles of affection and repulsion, like pendulum swings [00:21:46]. If your partner’s feelings are slowed by fear of vulnerability (analogous to friction), the system trends towards an inward spiral of mutual ambivalence [00:22:03].

This highlights that two vastly different systems (a physical pendulum and interpersonal dynamics) can share a similar underlying mathematical structure, especially evident in their phase diagrams [00:22:21]. The tactics developed for one case can often be transferred to many others [00:22:50].

Numerical Methods for Solving Differential Equations [00:23:00]

When exact analytic solutions are impossible, numerical methods for solving differential equations provide a way to approximate solutions [00:23:00].

Basic Idea: Finite Time Steps [00:23:05]

The core idea is to simulate the system by taking small, finite time steps. If you are at a point in the phase diagram, you take a step equal to delta_t (a small time increment) multiplied by the vector at that point [00:23:14]. Repeating this process for many small steps provides an approximation of the trajectory over time [00:23:34]. A smaller delta_t leads to a more accurate approximation but requires more steps [00:23:52].

Example: Python Program for Pendulum [00:24:19]

A simple Python program can numerically solve the pendulum equation:

1. Define initial theta and theta_dot (e.g., theta = pi/3, theta_dot = 0) [00:24:34].
2. Set a small delta_t (e.g., 0.01) [00:24:52].
3. Loop through time steps:
   • Increase theta by theta_dot * delta_t [00:24:58].
   • Compute theta_double_dot using the differential equation [00:25:02].
   • Increase theta_dot by theta_double_dot * delta_t [00:25:06].
4. Return the final theta value after all steps [00:25:11].

This is a basic form of numerical methods for solving differential equations, which can be made more sophisticated for better accuracy and efficiency [00:25:20].

Limitations and Chaos Theory [00:25:30]

Even with numerical solutions, there are inherent limits to how well we can use these systems for prediction [00:25:54]. Chaos theory, a major field that emerged in the last century, reveals that for some systems, tiny variations in initial conditions (due to imperfect measurements) can lead to wildly different trajectories [00:25:58]. The three-body problem, for example, contains elements of chaos [00:26:23].

While it seems “cruel” that the universe’s language is full of riddles that are either unsolvable or useless for long-term prediction, it also offers reassurance [00:26:28]. It suggests that the complexity observed in the world can be studied within the math itself, rather than being solely due to a mismatch between model and reality [00:26:45].