Differential Equations

A differential equation is an equation involving derivatives—it relates a function to its rates of change. Many real-world phenomena are naturally described by differential equations: how populations grow, how diseases spread, how radioactive substances decay, how electrical circuits respond to voltage changes. Rather than knowing the function directly, we know how it changes, and we must reverse-engineer the original function. This chapter introduces methods to solve differential equations, bridging the derivatives of chapter-06-application-of-derivatives and the integration of chapter-07-integrals.

Understanding Differential Equations

A differential equation involves a function and its derivatives. The order of a differential equation is the highest derivative present.

Examples:

• dy/dx = 2x is first-order (involves dy/dx only)
• d²y/dx² + 3dy/dx + 2y = 0 is second-order (involves d²y/dx²)

A solution to a differential equation is a function y = f(x) that satisfies the equation when substituted.

General vs. particular solutions:

• General solution: involves arbitrary constants (from chapter-07-integrals)
• Particular solution: specific solution satisfying initial conditions (e.g., y(0) = 5)

Example: For dy/dx = 2x,

• General solution: y = x² + C (any constant C)
• If y(0) = 3, then 3 = 0 + C, so C = 3
• Particular solution: y = x² + 3

Building from Earlier Chapters

chapter-05-continuity-and-differentiability introduced derivatives. chapter-07-integrals showed that integration recovers functions from derivatives. Now we systematically solve equations involving derivatives: given dy/dx in terms of x and y, find y.

Separable Differential Equations

A differential equation dy/dx = f(x, y) is separable if it can be written:

dy/dx = g(x) · h(y)

Solution method: Separate variables and integrate:

(1/h(y)) dy = g(x) dx
∫(1/h(y)) dy = ∫g(x) dx
F(y) = G(x) + C

Example: dy/dx = y

• Separate: (1/y) dy = dx
• Integrate: ∫(1/y) dy = ∫dx
• Result: ln|y| = x + C, so y = Ae^x (where A = ±e^C)

Example: dy/dx = (x/y)

• Separate: y dy = x dx
• Integrate: ∫y dy = ∫x dx
• Result: y²/2 = x²/2 + C, so y² = x² + K (where K = 2C)

Applications of Separable Equations

Exponential Growth and Decay

Many phenomena follow dy/dx = ky, where k is a constant:

• k > 0: exponential growth (bacteria, investments)
• k < 0: exponential decay (radioactive decay, cooling)

Solution: y = Ae^(kx) or y = y₀e^(kt) (with initial condition y(0) = y₀)

Example: A radioactive substance decays with dy/dt = -0.1y, where y is mass and t is time in years. If y(0) = 100 grams:

• Solution: y = 100e^(-0.1t)
• After 10 years: y = 100e^(-1) ≈ 36.8 grams

Newton's Law of Cooling

An object cools toward ambient temperature according to:

dT/dt = k(T - Tₐ)

where T is object temperature, Tₐ is ambient temperature, and k < 0.

Solution: T - Tₐ = Ae^(kt), so T = Tₐ + Ae^(kt)

With initial condition T(0) = T₀: A = T₀ - Tₐ

Linear Differential Equations

A first-order linear differential equation has the form:

dy/dx + P(x)y = Q(x)

Solution method: Use an integrating factor μ(x) = e^(∫P(x)dx). Multiply both sides by μ(x):

μ(x)dy/dx + μ(x)P(x)y = μ(x)Q(x)
d/dx[μ(x)y] = μ(x)Q(x)

Integrate both sides and solve for y.

Example: dy/dx - 2y = e^(3x)

• P(x) = -2, so μ(x) = e^(-2x)
• Multiply: e^(-2x)dy/dx - 2e^(-2x)y = e^x
• Recognize: d/dx[e^(-2x)y] = e^x
• Integrate: e^(-2x)y = e^x + C
• Result: y = e^(3x) + Ce^(2x)

Second-Order Linear Equations (brief introduction)

For equations like d²y/dx² + a(dy/dx) + by = f(x), solutions combine:

1. Complementary solution (homogeneous part): yc satisfying d²y/dx² + a(dy/dx) + by = 0
2. Particular solution (non-homogeneous part): yp

General solution: y = yc + yp

Connections to Other Topics

• chapter-06-application-of-derivatives: Differential equations model real-world rate problems
• chapter-07-integrals: Integration solves differential equations
• chapter-13-probability: Differential equations model stochastic processes
• Physics applications: motion, electricity, heat transfer

Socratic Questions

1. Explain why the general solution to a differential equation contains arbitrary constants. How do initial conditions pin down these constants to give a unique particular solution?

1. For the differential equation dy/dx = y, separate variables to get (1/y) dy = dx. Why is this valid as long as y ≠ 0? What happens if y = 0; is it a solution to the original equation?

1. Two populations follow dy/dt = ky and dz/dt = 3ky with the same k. If both start with the same initial value, how will the second population differ from the first? What does the factor of 3 on the right side change?

1. In separable equations like dy/dx = f(x)·g(y), why is separation of variables valid? That is, why can we rearrange to get (1/g(y)) dy = f(x) dx without loss of information?

1. For Newton's Law of Cooling, dT/dt = k(T - Tₐ), explain why the temperature difference (T - Tₐ) decays exponentially, not the absolute temperature T. How does this make intuitive sense?