W1. Direction Fields and Isoclines

1. Theory

A differential equation is an equation that relates an unknown function with one or more of its derivatives. Whenever a quantity changes — a falling body accelerates, a population grows, a cup of coffee cools — the law governing it connects the quantity to the rate of its change, and that connection is a differential equation. This chapter builds the language of the whole course: how applied problems become equations (1.1), how equations are classified (1.2), and how solutions can be seen geometrically without solving anything (1.3–1.4).

1.1 Applied Problems Leading to Differential Equations

Applying mathematics to a real problem means translating it into formulas — constructing a mathematical model. The model is always simpler than reality: for a falling object we ignore air resistance details and the pull of distant planets; for a population we pretend growth is continuous rather than birth-by-birth. A good model has exactly two properties: it is simple enough to be solvable, and faithful enough that its predictions match observations to a useful accuracy. If predictions disagree with experiment, the assumptions — not the mathematics — get revised.

Physics supplies the raw material: fluid motion, electric current, heat dissipation, seismic waves, population change. In each case some characteristic of the phenomenon is linked to its rate of change, and that link is an equation in derivatives.

• A word of history: the subject is as old as calculus. Newton founded classical mechanics in the Principia (1687); Leibniz, who published calculus first, gave us the term “differential equation” (1676) and the notation still used today.

Example 1: A Falling Object. A mass falls with gravity pulling down and air drag resisting, proportional to velocity with drag coefficient . Newton’s second law (force = mass × acceleration) gives: Dividing by produces a first-order ODE: Its solution is: with fixed by the initial velocity . As the exponential dies and velocity settles at the terminal velocity: * Worked check: for kg and kg/s, m/s — the same 49 m/s that will reappear as an equilibrium in 1.3.

Example 2: Population Dynamics (Malthusian Model). Malthus (1778) assumed births and deaths per capita constant: with birth coefficient and death coefficient , i.e. with net growth rate : Separation of variables yields exponential growth or decay: where . If the population explodes without bound; if it dies out; freezes it. The model fits hares without predators or bacteria in fresh broth — but finite food, space and disease eventually break it, which is why the logistic model comes later in the course. * Key Pitfall: exponential fit to early data says nothing about saturation; “no limiting factors” is the assumption doing all the work.

Example 3: Model of a Cooling Cup of Coffee. A body at cools toward ambient ; drinkable means roughly . Newton’s law of cooling states the heat lost in time : with surface area and heat transfer coefficient . The same heat leaves the internal energy, ( specific heat, density, volume): Dividing gives a first-order linear ODE: where is the cooling rate constant. The derivation assumes uniform body temperature (valid for small, conductive bodies) and no internal heat sources.

1.2 Basic Concepts and Classification of Differential Equations

1.2.1 Ordinary and Partial Differential Equations

Equations split by the number of independent variables. An ordinary differential equation (ODE) involves a function of one variable — e.g. the charge on a capacitor in an RLC circuit: A partial differential equation (PDE) involves several variables — e.g. the one-dimensional heat equation for temperature in a rod: with thermal diffusivity . This course studies ODEs; PDEs are a separate subject.

1.2.2 Order of a Differential Equation

The order is the highest derivative present. A general -th order ODE in implicit form: * Worked check: the Duffing equation is second order, because is its highest derivative — the term affects linearity, never order. A first-order ODE solved for its derivative (explicit/normal form) reads ; this is the form 1.3 and 1.4 will geometrize.

1.2.3 Systems of Differential Equations

One equation is often not enough: predators and prey, or coupled circuits, need several unknown functions at once. A system of first-order ODEs in normal form: where . If does not depend on explicitly, the system is autonomous — its future depends only on the current state, never on the clock. The Lotka–Volterra predator–prey model ( prey, predators) is an autonomous nonlinear system; its product terms already hint at the next classification.

1.2.4 Linear and Nonlinear ODEs

An ODE is linear if the unknown and its derivatives appear only to the first power, never multiplied or wrapped in nonlinear functions: The test is on and its derivatives only — the coefficients and may be arbitrary functions of . Anything else is nonlinear: the Duffing term, the Lotka–Volterra products , or an epidemic model coupling infected with susceptible through . * Why it matters: linearity decides the toolbox. Linear equations admit superposition and closed-form methods; nonlinear ones usually do not — which is exactly why the geometric methods of 1.3 exist.

1.2.5 Homogeneous and Nonhomogeneous ODEs

For a linear ODE, look at the right-hand side in (1.15). If every term contains or a derivative — i.e.  — the equation is homogeneous; a nonzero (external forcing, ambient temperature, voltage source) makes it nonhomogeneous. The zero function solves every homogeneous linear ODE and never a nonhomogeneous one. Cooling (1.10) is nonhomogeneous because of the constant term ; the RLC circuit (1.11) is nonhomogeneous exactly when the source ; the Lotka–Volterra system is homogeneous in the forcing sense — no external term. * Key Pitfall: “homogeneous” also has other meanings in mathematics (homogeneous functions, homogeneous systems in linear algebra). Here it strictly means “no -free term”.

1.3 Direction Fields for First-Order Equations

When formulas fail, geometry still works. For , calculus says the derivative at is the slope of the solution curve through that point. So at every point we already know the direction of the solution — without solving anything. A direction field (slope field) draws this knowledge: short segments of fixed length on a grid , each tilted with slope . Solution curves are the paths that thread through the segments tangentially; under standard conditions exactly one passes through each point.

Construction algorithm: pick a region and a grid; fix the segment length ; at each grid point compute ; draw the line and centre a length- segment on the grid point; repeat everywhere. * Key Pitfall: must stay smaller than the grid spacings , , or segments overlap into an unreadable blob. In practice nobody hand-draws hundreds of slopes — MATLAB/Python/Maple do — but reading the picture is a human skill.

Example (constant slope). For every segment has slope 1: the field is parallel lines, and the solutions are visible before any integration. The initial condition merely picks one line.

Example (equilibrium). For falling-body velocity : at the slope is (speeding up), at it is (slowing down), at it is — horizontal segments. The constant m/s is an equilibrium (steady-state) solution, and arrows on both sides point toward it: whatever the drop height, velocity converges to 49 m/s, matching (1.4).

Example (guessing formulas). For (away from the origin) each slope equals the slope of the ray from the origin — so integral curves must be the rays themselves. The field suggests the answer that integration would confirm.

Advantages: no analytic solution needed; behaviour visible at a glance; equilibria pop out; works for nonlinear equations; easy to read. Limitations: qualitative only, no numbers at specific points; accuracy lives and dies with grid density (a coarse grid misleads); hand-plotting is slow.

1.4 Isoclines

Evaluating at hundreds of grid points is expensive. An isocline (Greek isos, equal + klinein, to slope) groups the work: it is a curve along which the slope is constant. Draw a few such curves, hang equal-slope segments on each, and the field assembles itself.

1.4.1 Geometric Solution Algorithm Using Isoclines

1. Select representative slopes (e.g. ).
2. Plot each curve .
3. Draw short slope- segments along each curve.
4. Trace smooth integral curves from the initial point, hopping isocline to isocline while staying tangent to the segments; add more isoclines for accuracy.

Example (circles). For the isoclines are concentric circles of radius : is the origin (slope 0; note the ODE is not differentiable there), a unit circle of segments, slope-2 segments (). No square roots were ever computed — yet the integral curves curving up through the circles are immediate.

Geometric thinking — fields today, isoclines now — is the first step of the course’s qualitative toolkit: it builds the intuition that later selects solution methods, analyses stability and sanity-checks every model result.

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2. Definitions

• Mathematical model: A mathematical formulation of a physical or real-life problem, typically involving equations that relate an unknown function and one or more of its derivatives.
• Differential equation: An equation that contains an unknown function and one or more of its derivatives.
• Ordinary differential equation (ODE): An equation in which the unknown function depends on a single independent variable.
• Partial differential equation (PDE): An equation in which the unknown function depends on several independent variables.
• Order of a differential equation: The order of the highest derivative that appears in the equation.
• System of differential equations: A set involving several unknown functions and their derivatives.
• Autonomous system: A system of differential equations where the right-hand sides do not explicitly depend on the independent variable .
• Linear ODE: An equation where the dependent variable and its derivatives appear in a linear combination with coefficients depending only on the independent variable.
• Nonlinear ODE: A differential equation that is not linear with respect to the unknown function and its derivatives.
• Homogeneous linear ODE: A linear differential equation where every term is a function of the dependent variable or its derivatives, with no term independent of them.
• Nonhomogeneous linear ODE: A linear differential equation that includes an additional term independent of the unknown function and its derivatives.
• Direction field: A collection of short line segments or arrows drawn at grid points in the plane, showing the local slope of solution curves for a first-order ODE.
• Equilibrium solution: A constant solution of a differential equation where the derivative is identically zero, representing a steady-state condition.
• Isocline: A curve in the coordinate plane along which the slope of the direction field of a first-order differential equation is constant.

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3. Formulas

• Newton’s Second Law for a Falling Object: , valid for with mass , gravitational acceleration , and drag coefficient .
• Velocity of a Falling Object (Solution): , where is a constant determined by the initial condition .
• Terminal Velocity: .
• Malthusian Population Growth Model: , where is the net growth rate (with birth rate and death rate ).
• Population Growth Solution: , where is the initial population size at time .
• Newton’s Law of Cooling (Heat Transfer): , where is heat, is the heat transfer coefficient, is surface area, is temperature, and is ambient temperature.
• Internal Energy Change: , where is specific heat capacity, is density, and is volume.
• Differential Equation for Cooling: , where the cooling rate constant is .
• General -th Order ODE: .
• Explicit First-Order ODE: .
• System of First-Order ODEs (Normal Form): for .
• General Linear ODE of Order : , where coefficients and depend only on the independent variable .
• Linear Homogeneous ODE: .
• Isocline Equation: , where is a constant representing the slope of the direction field along the curve.

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4. Practice

4.1. Verifying Solution to Falling Object Equation (Lecture 1, Task 1)

Verify by substitution that the function is a solution to the first-order ordinary differential equation describing a falling object with air resistance:

Click to see the solution

Step 1: Compute the derivative of the proposed solution. Given the function: we differentiate both sides with respect to time :

Since is a constant, its derivative is . Using the chain rule for the exponential term:

Step 2: Substitute and into the left-hand side (LHS) of the differential equation. The left-hand side is :

Step 3: Substitute into the right-hand side (RHS) of the differential equation. The right-hand side is :

Distribute the through the parentheses:

Step 4: Compare LHS and RHS. Since , the given function satisfies the differential equation identically for any constant . Thus, the solution is verified.

Answer: LHS = RHS, so is a solution for any constant .

4.2. Malthusian Population Growth Model (Lecture 1, Task 2)

Solve the initial value problem for the Malthusian population model: where is a constant net growth rate.

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Step 1: Separate the variables. Rewrite the equation by moving all terms involving to the left-hand side and all terms involving to the right-hand side:

Step 2: Integrate both sides. Integrate the left side with respect to and the right side with respect to : where is an arbitrary constant of integration.

Step 3: Solve for . Exponentiate both sides to eliminate the natural logarithm:

Let (a new non-zero constant). We can write:

Step 4: Apply the initial condition. Use the initial condition to find the constant :

Substitute back into the general solution:

Thus, the unique solution to the initial value problem is:

Answer: .

4.3. Equilibrium Solution of a Falling Object (Lecture 1, Task 3)

Find the equilibrium (steady-state) velocity for an object falling under gravity with linear air resistance, governed by the differential equation: and determine its limit as for an arbitrary initial velocity .

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Step 1: Define the equilibrium condition. An equilibrium solution occurs when the state of the system does not change over time, meaning the time derivative is zero:

Step 2: Solve for the equilibrium velocity . Substitute into the differential equation:

Step 3: Analyze stability using the direction field properties. From the differential equation:

• If , then , so the velocity increases.
• If , then , so the velocity decreases.

Step 4: Determine the long-term limit. Because trajectories below increase toward it, and trajectories above decrease toward it, all solutions with any initial velocity approach the equilibrium value as time goes to infinity:

Answer: m/s, globally attracting.

4.4. Constructing Isoclines for a Nonlinear ODE (Lecture 1, Task 4)

Find and describe the isoclines for the first-order differential equation: and determine the slopes of the direction field along the isoclines for .

Click to see the solution

Step 1: Define the isocline equation. An isocline is a curve along which the slope is constant. Setting the right-hand side equal to a constant ():

Step 2: Simplify the equation. Square both sides of the equation: This is the equation of a family of concentric circles centered at the origin with radius equal to .

Step 3: Evaluate specific slope values. * For : , which yields a single degenerate point at the origin . The slope of the direction field at this point is . * For : , which is a circle of radius . At every point on this circle, all direction field segments have a constant slope of (an angle of ). * For : , which is a circle of radius . At every point on this circle, all direction field segments have a constant slope of (an angle of ).

Answer: Isoclines are concentric circles with slopes , , .