Teacher page: first order differential equations (AA 5.18)

The one thing to do with the figure

Make them classify before they solve anything.

Put the four equations on the board with no method attached and ask which is which. The chooser then confirms it and shows the working.

Nearly all the marks lost in this sub-topic are lost to a wrong choice, not to bad algebra, so the classification deserves the lesson time.

The ten second homogeneity test

Replace x with λx and y with λy. If every λ cancels, it is homogeneous and y = vx will work.

Teach it as a reflex. Without it students attempt separation on equations that cannot separate and lose the question before starting.

The answers

Separable	dy/dx = xy with y(0) = 1 gives y = e^x²/2, so y(2) = e² ≈ 7.389.
Homogeneous	dy/dx = (x+y)/x with y = vx gives x dv/dx = 1, so y = x(ln x + C).
Integrating factor	y′ + y/x = x has IF = x, giving y = x²/3 + C/x.
Euler	dy/dx = x + y, y(0) = 1, h = 0.1 reaches 1.72102 at x = 0.5; exact 1.7974.
1, 2, 3	B, integrating factor; 7.389; 1.22.

Where the marks go

1 markIdentifying the method, with a reason.

1 markSetting it up: the substitution, or the integrating factor simplified.

1 markIntegrating both sides correctly.

1 markUsing the initial condition.

What each wrong answer tells you

They give	What it means
Separating y′ + y/x = x	Cannot be done, because y appears added rather than multiplied. Their first instinct, every time.
y = vx without the product rule	dy/dx = v + x dv/dx, not just v. The missing term stops it working.
IF left as e^{ln x}	Correct but unsimplified, and the next line is much harder than it needs to be.
Rounding in the Euler table	The errors compound. Keep full accuracy, round once.

Other things they will say

"How do I know it is homogeneous?" The λ test, in ten seconds. Do not guess from the look of it.

"Where does the integrating factor come from?" It is engineered so the left side becomes a product derivative. Show that happening once; it stops being arbitrary.

"Can one equation take two methods?" Often, yes. dy/dx = x + y works by integrating factor and by Euler. Use whichever the question asks for.

A possible order

	What is happening
1	Four equations, no methods. Classify them as a class.
2	Separable and homogeneous, with the λ test.
3	The integrating factor, including why it works.
4	Euler, as a table.
5	Questions 1 to 3.
6	A mixed set where the method is not stated.

Two things not to say

Do not set exercises grouped by method. Students then never practise choosing, which is the part that is assessed.

Do not skip the derivation of the integrating factor. Presented as a formula it is forgotten within a week.

Questions to set

Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.

1Fluency

The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.

SeparableSolve dy/dx = y/x with y(1) = 2.
ln y = ln x + c gives y = 2x. At x = 5, y = 10.
Check itVerify that solution satisfies the equation.
dy/dx = 2 and y/x = 2x/x = 2. They agree for all x, not just at the initial point.

2In context

The same skill inside a real situation, where the first job is working out what is being asked.

Integrating factorSolve dy/dx + y = x with y(0) = 0.
The factor is e^x, giving y = x − 1 + Ce^−x. With y(0) = 0, C = 1, so y = x − 1 + e^−x. At x = 1, y = 1/e = 0.368.
Logistic equilibriaFor dP/dt = 0.1P(1 − P/1000), find the equilibrium values and say which is stable.
P = 0 and P = 1000. P = 1000 is stable, since the rate is positive below it and negative above; P = 0 is unstable.

3Challenge

Reasoning, working backwards, or spotting an error. These are where the top grades are decided.

Name the method from the shapeFor dy/dx = xy, dy/dx + 2y = e^x and dy/dx = x + y, state which method each needs.
Separable, integrating factor, integrating factor. Separation works only when the equation factorises into a function of x times a function of y, and dy/dx = x + y does not, which is the one students try hardest to separate.
Why the constant mattersExplain why a general solution with an arbitrary constant is not an answer to an initial value problem.
The constant is a whole family of curves filling the plane. The initial condition selects the single member passing through the given point. An answer left with C in it has not answered the question asked.

Practicalities

Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.