Porisma

Differential equations

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

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.

Open the chapter exploration

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

35 examples and practice questions, in 6 sections.

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.

Ready to try this on your own?

For “Differential equations”, try a question without the working and explain your method. This records your confidence, saved in this browser only.

Ready to practise?Chapter practice