Differential equations

Since Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations, Steven Strogatz

The real problem of humanity is the following: We have Paleolithic emotions, medieval institutions and godlike technology. And it is terrifically dangerous, and it is now approaching a point of crisis overall, Edward O. Wilson.

The problem is not the problem. The problem is your attitude about the problem, Captain Jack Sparrow.

Recall: Antiderivatives and Integration

Antiderivatives are one of the most fundamental ideas in calculus. They represent the inverse operation of differentiation.

Definition. Given a function f(x), an antiderivative, also called an indefinite integral, F is the function whose derivative gives back the original function, that is, F’ = f, e.g., 3x² -1 is the antiderivative of x³ -x +7. Symbolically, we write this as F(x) = $\int f(x)dx$.

Because the derivative of a constant is zero, antiderivatives are not unique. If F(x) is one antiderivative, then so is F(x) + C, where C is any constant. Thus, we usually write: $\int f(x)\,dx = F(x) + C.$

The process of finding antiderivatives is called integration.  

Differential equations

An understanding of differential equations is essential in many fields, including physics (motion, waves, heat flow), engineering (circuits, mechanics), economics (growth models), biology (population growth), and more. They are powerful tools for modeling and analyzing systems where change is a fundamental aspect.

Algebraic Equations

An algebraic equation is a mathematical statement asserting that two algebraic expressions are equal.. These expressions are constructed using:

1. Dependent and independent variables. Variables are symbols representing unknown quantities. Variables can be:
   Independent Variables: Variables or input that can be chosen (or change) freely.
   Dependent Variables: Variables that depend on the independent variables.
2. Constants. Fixed numerical values that do not change, e.g., 2, π, 7.
3. Algebraic operations. Operations such as addition, subtraction, multiplication, division, exponentiation, powers, and root.

An algebraic equation typically has the following general form: Expression₁ = Expression₂, where the two expressions are set equal to each other.

Examples: y = 2x + 4 (simple linear equation, represents a straight line with a slope of 2 and a y-intercept of 4), ax² + bx + c = 0 (quadratic equation, it represents a parabola), $ln(xy + t) = z·sin(\sqrt{x})$.

Analytical methods for solving various types of equations

Understanding algebraic equations is fundamental before tackling differential equations. Here are some common types:

• Linear Equations. Form: y = ax+b. Solution: This is a straight line where a is the slope (it measures the steepness) and b is the y-intercept (the point where the line crosses the y-axis). To find the point where the line intersects the x-axis, set y = 0: 0 = ax +b⇒ $x = \frac{-b}{a}$.
• Quadratic Equations. Form: $y = ax^2 +bx +c$. Solution: This is a parabola. The x-intercepts (roots) can be found using the quadratic formula: $x = \frac{-b±\sqrt{b^2-4ac}}{2a}$. Discriminant: Δ = b² -4ac. If Δ > 0: Two real and distinct roots; If Δ = 0: One real root (repeated); If Δ < 0: Two complex conjugate roots.
• Cubic Equations. Form: $y = ax^3 +bx^2 +cx +d$. The general solution involves Cardano’s method for finding real roots.
• Exponential Equations. Form $y = ae^{bx}↭\frac{y}{a} = e^{bx}↭ ln(\frac{y}{a}) = bx ↭ \boxed{x = \frac{ln(\frac{y}{a})}{b}}$

Differential Equations

Differential equations describe how quantities change over time or space. They are essential for modeling dynamic systems — from population growth and heat flow to electrical circuits, planetary motion, and economic models — where variables depend on one another.

Definition. A differential equation is an equation that involves one or more functions and their derivatives. It relates the function itself (the dependent variable), its derivatives with respect to one or more independent variables (rates of change), and the independent variable(s) themselves, e.g., $\frac{dy}{dx} = 3x +5y, y' + y = 4xcos(2x), \frac{dy}{dx} = x^2y+y, etc.$

If an algebraic equation is a “snapshot” of a relationship, a differential equation is the “instruction manual” for how that relationship moves and changes.

Key Components (e.g., $\frac{dy}{dx} = 3x +5y$):

• Dependent variables: The function y (or y(x)) we are trying to find or solve for. Variable(s) (y) that depend on one or more other variables.
• Independent variables: Variable(s) (x) upon which the dependent variables depend and we differentiate with respect to (often time or space).
• Derivatives: Expressions like $\frac{dy}{dx}$ or y’ that represent the rate at which the dependent variable (y) change with respect to the independent variable (x).

Famous Examples of Differential Equations

Differential equations serve as models for numerous phenomena in science, engineering, and everyday life. Here are some famous examples that illustrate the diversity and application of differential equations:

• Exponential Growth and Decay: $\frac{dy}{dt} = ky$. Solution: $y(t)=y_0e^{kt}$. When k > 0, we observe exponential growth (e.g., population growth or compound interest). The rate of change is proportional to the current amount. When k < 0, we have exponential decay (e.g., radioactive decay or cooling).
• Newton’s Law of Cooling: $\frac{dT}{dt} = -k(T-T_{env})$. The rate at which an object cools is proportional to the difference between its temperature and the room temperature. Solution: $T(t) = T_{env} + (T_0 -T_{env})e^{-kt}$. This equation describes how an object’s temperature T changes over time, approaching the surrounding environmental temperature T_env exponentially. This model applies to cooling or heating processes.
• Simple Harmonic Motion: $\frac{d^2y}{dt^2}+w^2y = 0$. Solution: y(t) = Acos⁡(ωt) + Bsin⁡(ωt). This describes oscillating systems, such as a pendulum or a mass on a spring. The solution represents periodic motion with angular frequency ω, where A and B depend on initial conditions.

Classification of Differential Equations

Differential equations are classified based on several criteria, each providing insight into the structure and solutions of the equation. Here is an expanded guide to these classifications:

1. Order: The order of a differential equation is the order of the highest derivative present in the equation. First-order differential equation: Involves only the first derivative ($\frac{dy}{dx}$, e.g. y′=2(25−y), y’ + 2x = sin(y), y’ -e^xy = 0). Second-order differential equation: Involves up to the second derivative ($\frac{d^2y}{dx^2}$, e.g., y’ + y’’ = 2x, $\frac{d^2y}{dx^2} + 4\frac{dy}{dx} +3y = 0$.)
2. Degree. The degree of a differential equation refers to the exponent of the highest derivative, assuming the equation has been made free from radicals and fractions involving derivatives, e.g., (y’’)² + y’ = ln(x). Here, the highest derivative y′′ is squared, giving the equation a degree of 2.
3. Linearity: A differential equation is linear if it can be written in the form: $a_n(x)\frac{d^ny}{dx^n}+ a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}} + ··· + a_0(x)y= g(x)$ where a_n(x), a_n-1(x), ···, a₀(x) and g(x) are function of the independent variable x, and the dependent variable y and all its derivatives appear linearly (i.e., no powers -only to the first power- or products of y and its derivatives) e.g., y’ + sin(x)y = e^x, y’’ + 3y’ + 2y = x². It is nonlinear if it involves nonlinear terms of the dependent variable or its derivatives, e.g., y·y’ + 2x = 8 (product of $y$ and its derivative) , (y’’)² + y’ = ln(x), sin(y) + y’ = 3 (trigonometric function of the dependent variable).
4. Ordinary Differential Equations (ODEs) vs. Partial Differential Equations (PDEs)

• Ordinary differential equations or (ODEs) are equations where the derivatives are taken with respect to a single independent variable, e.g., $\frac{dy}{dx} = 5x + y$ (only $x$ is the independent variable); $\frac{d^2y}{dx^2}+\frac{dy}{dx} = 3xsin(y)$. This equation involves derivatives with respect to x only, so it’s an ODE;
• Partial differential equations or (PDEs) are equations where derivatives are taken with respect to multiple independent variables., e.g., $\frac{∂y}{∂t}+z\frac{∂y}{∂z} = \frac{x-t}{x+t}$. This equation involves partial derivatives with respect to both t and z, making it a PDE.

$\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$. The Heat Equation: temperature $u$ depends on both time $t$ and position $x$.

A partial differential equation (PDE) is a type of mathematical equation that involves a function of several variables and its partial derivatives with respect to those variables, e.g., $\frac{∂f}{∂t} = \frac{∂^2f}{∂x^2}, \frac{∂w}{∂t}-\frac{∂^2w}{∂x^2} = 0, etc.$ Partial differential equations are incredibly important in many fields of science and engineering.

Solving First-Order Ordinary Differential Equations

The simplest first-order ODE is of the form $\frac{dy}{dx} = f(x)$. To solve this, we use Direct Integration.

This type of equation represents a first-order ordinary differential equation (ODE) because it involves only the first derivative of y with respect to x, and the rate of change of y with respect to x depends only on x. The goal is to find the function y(x) that satisfies this equation.

Steps to Solve the Equation

1. Isolate all terms involving y on one side and all terms involving x on the other side by multiplying both sides by dx: dy = f(x)dx. Now, all y terms are on the left, and all x terms are on the right.
2. Integrate both sides with respect to their respective variables: $\int dy = \int f(x)dx ↭[\text{This yields}] y + C_1 = F(x) + C_2$ where C₁ and C₂ are constants of integration. Combining them into a single constant C = C₂ -C₁, simplifies the expression.
3. Write down the general solution. The general solution for y(x) is y = $\int f(x)dx + C = F(x) + C$ where F(x) is the antiderivate of f(x) and C represents the constant of integration, accounting for all possible vertical shifts of the solution curve, giving a family of solutions.

Examples of Solving First-Order ODEs

• y’ = x²e^x⇒[Integrate both sides] $y = \int x^2e^xdx + C$ [Integration by parts ∫udv=uv−∫vdu, u = x² ⇒ du = 2xdx; dv = e^xdx ⇒ v = e^x] = $x^2e^x -\int 2xe^xdx$ [Integration by parts ∫udv=uv−∫vdu, u = 2x ⇒ du = 2dx; dv = e^xdx ⇒ v = e^x] = $x^2e^x -(2xe^x-\int 2e^xdx) = x^2e^x-2xe^x +2e^x + C$ where C has been renamed conveniently for simplicity.

• $y' = \frac{x^3}{\sqrt{1-x^2}} ⇒[\text{Integrate Both Sides:}] y = \int \frac{x^3}{\sqrt{1-x^2}}dx$ [We choose the trigonometric substitution x = sin(θ), by the Pythagorean Theorem sin²(θ)+cos²(θ) = 1 ⇒ 1 -sin²(θ) = cos²(θ) ⇒ $\sqrt{1-x^2}=cos(θ)$. x = sin(θ), then differentiating both sides gives: dx = cos(θ)dθ] $\int \frac{sin^3(θ)cos(θ)dθ}{cos(θ)}dx = \int sin^3(θ)dθ = \int sin^2(θ)sin(θ)dθ = \int (1-cos^2(θ))sin(θ)dθ$ [Substitution u = cos(θ), du = -sin(θ)dθ] $\int (u^2-1)du = \frac{1}{3}u^3-u + C = \frac{1}{3}cos^3(θ)-cos(θ)+C = \frac{1}{3}(1-x^2)^{\frac{3}{2}}-\sqrt{1-x^2}+C$

  The solution to a first-order ODE generally represents a family of curves, each differing by a constant of integration C.

To determine a specific solution, we need an initial condition, such as: $y(x_0) = y_0.$

Big Picture

• Algebraic equations describe static relationships — a snapshot in time, e.g., $2x + 3y = 12$.

• Differential equations describe change and dynamic behavior — how things evolve over time or space.

• Antiderivatives allow us to reverse differentiation — turning rates of change back into accumulated quantities. Your speedometer shows instantaneous speed (derivative). The odometer (total distance traveled) is the antiderivative (integral) of speed over time.

• Solving a simple first-order ordinary differential equation ODE is fundamentally about finding an antiderivative (plus a constant).

In many ways, integration is the bridge between rates of change and total accumulated behavior — and differential equations are the language of that bridge.

Bibliography

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2. Algebra, Second Edition, by Michael Artin.
3. LibreTexts, Calculus. Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
4. Field and Galois Theory, by Patrick Morandi. Springer.
5. Michael Penn, and MathMajor.
6. Contemporary Abstract Algebra, Joseph, A. Gallian.
7. YouTube’s Andrew Misseldine: Calculus. College Algebra and Abstract Algebra.
8. MIT OpenCourseWare, 18.01 Single Variable Calculus, Fall 2007M. 18.02 Multivariable Calculus, Fall 2007. 18.03 Differential Equations, Spring 2006.
9. Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences.