Differential Equations

A differential equation (DE) is any equation that relates a function $y(x)$ to one or more of its derivatives. The unknown is the function itself — you're solving for $y$, not a number.

The order of a DE is the highest derivative that appears. First-order: $y' = 2xy$ (involves $y'$ but not $y''$). Second-order: $y'' + 4y = 0$ (involves $y''$). Higher-order DEs exist but are less common in a first course.

A solution is a function $y(x)$ that, when substituted into the equation, makes it true for all $x$ in some interval. To verify a solution, plug it in and check: does both sides of the equation agree?

The general solution contains arbitrary constants: one constant for a first-order DE, two for second-order, etc. These constants represent the 'degrees of freedom' — infinitely many functions satisfy the DE, differing only in these constants. An initial value problem (IVP) pins down these constants by specifying conditions like $y(0) = 3$.

Linear vs. nonlinear: a DE is linear if $y$ and its derivatives appear to the first power with no products like $y \cdot y'$. Linear DEs have well-developed solution theory; nonlinear DEs are harder and may not have closed-form solutions.

A direction field visualizes a first-order DE $y' = f(x,y)$ by drawing short line segments with slope $f(x,y)$ at many points $(x,y)$.

The solution curves must be tangent to these segments at every point. So by drawing segments, you can see the shape of solutions without solving the equation.

Direction fields build intuition: you can spot equilibrium solutions (horizontal segments), identify whether solutions grow or decay, and see the qualitative behavior before doing any algebra.

Example: for $y' = -y$, the slopes are negative when $y > 0$ and positive when $y < 0$. Solutions decay toward $y = 0$, which matches $y = Ce^{-x}$.

A first-order DE is separable if it can be written as $\frac{dy}{dx} = g(x) \cdot h(y)$, where one factor depends only on $x$ and the other only on $y$.

To solve: separate the variables: $\frac{1}{h(y)}\,dy = g(x)\,dx$. Then integrate both sides.

Don't forget the constant of integration! Usually written as $+C$ on one side.

Example: $\frac{dy}{dx} = xy$. Separate: $\frac{1}{y}\,dy = x\,dx$. Integrate: $\ln|y| = \frac{x^2}{2} + C$. Solve: $y = Ae^{x^2/2}$ where $A = \pm e^C$.

Watch for lost solutions: when dividing by $h(y)$, any value where $h(y) = 0$ might be a constant solution (equilibrium) that you lose in the algebra.

The DE $\frac{dy}{dt} = ky$ says: the rate of change of $y$ is proportional to $y$ itself. It's separable: $\frac{dy}{y} = k\,dt$, integrating gives $\ln|y| = kt + C$, so $y(t) = y_0 e^{kt}$ where $y_0 = y(0)$.

If $k > 0$: exponential growth. The larger $y$ gets, the faster it grows. Applications: population growth (early phase), compound interest, viral spread, chain reactions.

If $k < 0$: exponential decay. The quantity shrinks at a rate proportional to its current size. Applications: radioactive decay, drug elimination from the body, cooling (simplified), depreciation.

Half-life: the time for the quantity to halve. Set $y_0 e^{kt_{1/2}} = y_0/2$, giving $t_{1/2} = \frac{\ln 2}{|k|}$. This is constant — it doesn't depend on the current amount. Carbon-14 has a half-life of ~5730 years regardless of how much you start with.

Doubling time: for growth ($k > 0$), the time for the quantity to double is $t_d = \frac{\ln 2}{k}$. The 'Rule of 70': at $r\%$ growth per year, the doubling time is approximately $70/r$ years.

Newton's Law of Cooling states that an object's temperature changes at a rate proportional to the difference between its temperature and the surrounding (ambient) temperature: $\frac{dT}{dt} = -k(T - T_{\text{env}})$, where $k > 0$.

This is separable. Let $u = T - T_{\text{env}}$, then $du/dt = -ku$, giving $u = u_0 e^{-kt}$. Substituting back: $T(t) = T_{\text{env}} + (T_0 - T_{\text{env}})e^{-kt}$.

Behavior: as $t \to \infty$, $T(t) \to T_{\text{env}}$. The object approaches room temperature exponentially — quickly at first (when the temperature gap is large), then slower as it gets closer.

The constant $k$ depends on the object's material, surface area, and the surrounding medium (air vs. water, for instance). A larger $k$ means faster cooling. Determining $k$ usually requires two temperature readings at known times.

Applications: forensic science (estimating time of death from body temperature), cooking (how long until food cools to eating temperature), engineering (heat dissipation in electronics).

Standard form: $\frac{dy}{dx} + P(x)y = Q(x)$. The key: this is always solvable using an integrating factor.

Integrating factor: $\mu(x) = e^{\int P(x)\,dx}$.

Multiply the entire equation by $\mu$: the left side becomes $\frac{d}{dx}[\mu(x) \cdot y]$.

Integrate both sides: $\mu(x) \cdot y = \int \mu(x) Q(x)\,dx + C$.

Solve for $y$: $y = \frac{1}{\mu(x)}\left[\int \mu(x) Q(x)\,dx + C\right]$.

Why it works: multiplying by $\mu$ turns the left side into a perfect derivative, which can be integrated directly.

The logistic equation models growth with a carrying capacity $K$: $\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$.

When $P$ is small compared to $K$, growth is approximately exponential ($\approx rP$). As $P$ approaches $K$, growth slows and stops.

Solution: $P(t) = \frac{K}{1 + Ae^{-rt}}$ where $A = \frac{K - P_0}{P_0}$.

The graph is an S-shaped (sigmoidal) curve. The population grows fastest at $P = K/2$ (the inflection point).

Applications: population biology, spread of diseases, adoption of technology, logistic regression in machine learning.

A Bernoulli equation has the form $\frac{dy}{dx} + P(x)y = Q(x)y^n$ where $n \neq 0, 1$. When $n = 0$ or $n = 1$, the equation is already linear. For other values of $n$, the $y^n$ term makes it nonlinear.

The clever trick: substitute $v = y^{1-n}$. Then $\frac{dv}{dx} = (1-n)y^{-n}\frac{dy}{dx}$. Divide the original Bernoulli equation by $y^n$, and the substitution transforms it into a linear first-order equation in $v$: $\frac{dv}{dx} + (1-n)P(x)v = (1-n)Q(x)$.

Solve this linear equation using the integrating factor method from the earlier section. Once you have $v(x)$, convert back: $y = v^{1/(1-n)}$.

Example: $y' + y = y^2$ is Bernoulli with $n = 2$. Let $v = y^{1-2} = y^{-1} = 1/y$. Then $v' = -y^{-2}y'$. Dividing by $y^2$: $y^{-2}y' + y^{-1} = 1$, which becomes $-v' + v = 1$, or $v' - v = -1$. This is linear and solvable with an integrating factor.

Bernoulli equations appear in population models with harvesting, certain fluid dynamics problems, and logistics applications.

Standard form: $ay'' + by' + cy = 0$ with constant coefficients.

Guess $y = e^{rx}$, substitute, and get the characteristic equation: $ar^2 + br + c = 0$.

Case 1 - Two distinct real roots $r_1, r_2$: general solution is $y = C_1 e^{r_1 x} + C_2 e^{r_2 x}$.

Case 2 - Repeated root $r$: general solution is $y = (C_1 + C_2 x)e^{rx}$. The extra $x$ factor accounts for the repeated root.

Case 3 - Complex roots $r = \alpha \pm \beta i$: general solution is $y = e^{\alpha x}(C_1 \cos(\beta x) + C_2 \sin(\beta x))$. This produces oscillatory behavior (springs, circuits).

Form: $ay'' + by' + cy = g(x)$ where $g(x) \neq 0$. The right-hand side $g(x)$ represents an external 'forcing' — a driving input that pushes the system.

The general solution has two parts: $y = y_h + y_p$. Here $y_h$ is the general solution to the homogeneous equation ($g = 0$) — it captures the system's natural behavior. $y_p$ is any particular solution that accounts for the forcing. The combination gives every possible solution.

Method of undetermined coefficients: guess the form of $y_p$ based on $g(x)$ and solve for the unknown coefficients. The guesses follow a pattern:

If $g(x) = Ce^{kx}$: try $y_p = Ae^{kx}$. If $g(x) = C_n x^n + \cdots + C_0$ (polynomial): try $y_p = A_n x^n + \cdots + A_0$ (same degree). If $g(x)$ involves $\sin(kx)$ or $\cos(kx)$: try $y_p = A\cos(kx) + B\sin(kx)$ (always include both sin and cos). If $g$ is a combination, use superposition — try the sum of individual guesses.

Duplication rule: if your guess for $y_p$ duplicates a term in $y_h$, multiply by $x$ (or $x^2$ if it duplicates a repeated-root term). This ensures $y_p$ is genuinely new and not absorbed into $y_h$.

Variation of parameters: a more general method that works for any $g(x)$, not just the special forms above. It uses $y_p = u_1 y_1 + u_2 y_2$ where $y_1, y_2$ are the homogeneous solutions and $u_1, u_2$ are functions found via integrals. More work, but universally applicable.

A mass-spring system obeys $my'' + cy' + ky = F(t)$ where $m$ is mass, $c$ is damping, $k$ is spring constant, and $F(t)$ is external force.

Undamped free oscillation ($c=0$, $F=0$): $y'' + \omega^2 y = 0$ with $\omega = \sqrt{k/m}$. Solution: $y = A\cos(\omega t) + B\sin(\omega t)$. Pure sinusoidal motion.

Damped oscillation ($c > 0$, $F=0$): the solution includes $e^{\alpha t}$ with $\alpha < 0$, causing the amplitude to decay over time.

Overdamped ($c^2 > 4mk$): no oscillation, just slow return to equilibrium.

Critically damped ($c^2 = 4mk$): fastest return to equilibrium without oscillating.

Underdamped ($c^2 < 4mk$): oscillation with decaying amplitude.

An IVP provides the DE plus enough initial conditions to pin down a unique solution from the infinite family of general solutions.

First-order DE: one initial condition, typically $y(x_0) = y_0$. This determines the one arbitrary constant $C$ in the general solution.

Second-order DE: two initial conditions, typically $y(x_0) = y_0$ (starting position) and $y'(x_0) = v_0$ (starting velocity). These determine the two constants $C_1$ and $C_2$.

Procedure: (1) find the general solution (with arbitrary constants), (2) apply the initial conditions to get a system of equations for the constants, (3) solve the system, (4) write the particular solution with specific constants.

The Existence and Uniqueness Theorem (Picard-Lindelöf): if $f(x,y)$ is continuous and satisfies a Lipschitz condition in $y$ near $(x_0, y_0)$, then the IVP $y' = f(x,y)$, $y(x_0) = y_0$ has exactly one solution in some interval around $x_0$. This guarantees that solutions don't 'branch' or 'disappear' — there's always exactly one curve through each initial point.

Physical interpretation: knowing the DE gives you the rules of the system (how things change). The initial conditions specify the starting state. Together, they completely determine the future (and past) behavior. This is why differential equations are the language of deterministic physics.