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Differential equations

Integrate and separate first-order equations, use substitutions, and interpret rate models with their initial conditions.

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By the end, you can

  • Separate variables while preserving equilibrium solutions
  • Use conditions to find particular solutions
  • Interpret growth and decay models in context

about 20 minutes for the lesson · practise at your own pace.

The central idea

A differential equation specifies a rate in terms of the changing quantities. If dy/dx=f(x)g(y)dy/dx=f(x)g(y), separate variables where g(y) is nonzero and integrate. Values with g(y)=0 may give equilibrium solutions and must be checked separately. A condition then selects one member of the solution family.

Order, linearity and initial conditions

A differential equation relates an unknown function and derivatives; its order is the highest derivative. An ordinary differential equation has one independent variable. A linear one has form ∑k=0nfk(x)y(k)=g(x)\sum_{k=0}^n f_k(x)y^{(k)}=g(x), with coefficients depending only on x.

EquationOrderLinear?
x+3 dy/dx=4yx+3\,dy/dx=4y1Yes
dy/dx=sin⁡ydy/dx=\sin y1No
d2y/dx2+3 dy/dx+4y=x2d^2y/dx^2+3\,dy/dx+4y=x^22Yes
d3y/dx3+2(dy/dx)2+5y=sin⁡xd^3y/dx^3+2(dy/dx)^2+5y=\sin x3No

Verify a solution by substituting it and its derivatives. For dy/dx=2xdy/dx=2x, the general family is y=x2+Cy=x^2+C. If y=10y=10 at x=5x=5, C=−15C=-15 and the particular solution is y=x2−15y=x^2-15. Conditions select constants. The source’s usual n-constant description presumes a regular nth-order problem; singular solutions can require separate consideration. The course focus is first-order equations.

Explore

Consider dV/dt=6−0.2VdV/dt=6-0.2V, where VV is in litres and tt in minutes. Compare initial volumes 0, 20 and 40 L. Predict each initial rate, whether the volume rises or falls, and any common limiting volume before solving. Then solve each initial-value problem and check the predictions. State the units of 6 and 0.2, and explain why changing the initial volume changes the particular solution without changing the balance law.

Related interactive example

Explore this related example, then return to the prediction task above.

The family y = ln(x² + 1) + C

Each dash shows the slope the differential equation gives at that point. Change the starting value and the solution follows the dashes.

Starting value of y at x = 00
Read the solution at x2
Try an example
y at x = 2
1.609
Rate dy/dx there
0.8
  • Solution that starts at the purple dot
  • Closed-form answer
Key idea

All solutions are copies of ln⁡(1+x2)\ln(1+x^2) shifted vertically. The particular one through the origin has constant C=0C=0.

Ideas to try
  1. Move the starting value up and down: the curves are vertical shifts of each other.
  2. Read the gradient at x=1x=1 and at x=4x=4.
Open the full interactive problem →

Worked example

An object cools in a room held at 20∘C20^\circ\mathrm C. Assume its rate of temperature change is proportional to its difference from room temperature, with constant k>0k>0. Then dT/dt=−k(T−20)dT/dt=-k(T-20).

If T(0)=80T(0)=80, separation and the initial condition give T=20+60e−ktT=20+60e^{-kt}.

If T(10)=50T(10)=50, then e−10k=1/2e^{-10k}=1/2, so k=ln⁡2/10 min−1k=\ln2/10\ \mathrm{min}^{-1}. Hence T=20+60 2−t/10T=20+60\,2^{-t/10}.

The object reaches 35∘C35^\circ\mathrm C at t=20t=20 minutes and approaches 20∘C20^\circ\mathrm C, rather than zero, as time increases.

Substitute the solution into the rate equation and check both observations.

The quantity that decays here is the difference from room temperature. Check the constant solution before dividing by that difference, and preserve its sign until the initial condition determines it.

Check your understanding

A 60 L tank initially contains 8 L of liquid. Liquid enters at 8 L/min and drains at a rate proportional to the current volume: one quarter of the volume per minute.

Assume these rates continue unchanged.

Form and solve a differential equation for the volume.

Find when it first reaches 20 L, and decide whether the model predicts that the tank fills to capacity.

Check the initial rate and the units of the proportionality constant.

Mastery learning objectives

At the end of this chapter, I should be able to:

Differential Equations: after lecture and after tutorial

Solve dy/dx=f(x)dy/dx=f(x) by direct integration.

Solve dy/dx=g(y)dy/dx=g(y) by separation of variables, checking values excluded by division.

Solve dy/dx=f(x)g(y)dy/dx=f(x)g(y) by separation of variables.

Use a given substitution to reduce an equation to direct integration or separation, then restore the original variables.

Choose a practice question

Common question typeLecture examplesDiscussionReview follow-up
Direct integration1, 211, 5(a)
Separation of variables3, 4, 1358
Substitution5, 6, 72, 3, 43
Applications8, 9, 10, 11, 126, 7, 8, 9, 102, 4, 5(b), 6, 7, 9, 10

The printed map places Review1 under direct integration although its loan equation is separable, and Review2/4 under applications although they are abstract differential-equation exercises. These source routes are retained rather than silently corrected; the canonical working identifies the actual method. Lecture Example13 is source enrichment as described in its working.

Practice questions

Practice section 1 · Questions 1–6

Integrate a logarithmic derivative directly · Apply initial data and an inverse-square gradient law · +4 more

Practice section 2 · Questions 7–12

Differentiate a substitution before replacing every dependent term · Use the product rule in a homogeneous-looking substitution · +4 more

Practice section 3 · Questions 13–18

Use half-life data in a signed exponential decay model · Keep both sides of a terminal velocity when separating · +4 more

Practice section 4 · Questions 19–24

Solve a homogeneous first-order differential equation · Separate variables and integrate by parts · +4 more

Practice section 5 · Questions 25–30

Solve logistic growth and sketch its changing curvature · Separate continuous payoff time from monthly rounding · +4 more

Practice section 6 · Questions 31–35

Derive and calibrate a logistic population model · Derive the insect model and restore its zero equilibrium · +3 more

More practice

Build the rate equation from the stated mechanism before solving it. Check constants, initial conditions, units and long-run behaviour against the situation.

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