Differential equations and separation of variables

Higher Level

"The population grows at a rate proportional to its size" is dP/dt = kP. Every faint curve below satisfies it. Move the starting value and watch one of them get picked out.

P(0) = 100

k = 0.20

P = 100e^0.2tthe solution

271.8P when t = 5

One equation, infinitely many curves. The measurement at t = 0 chooses one.

Reading the sentence

The words	The equation
grows at a rate proportional to its size	dP/dt = kP
cools at a rate proportional to the temperature difference	dT/dt = −k(T − R)
the rate of decay is proportional to the amount left	dm/dt = −km

"Proportional to" means = k ×, and the k is unknown until the question gives you enough to find it. A decreasing quantity carries a minus sign, either in front or inside k, but say which.

Separating the variables

Get every y with the dy and every x with the dx, then integrate both sides. It works whenever the right-hand side is a function of x times a function of y.

Worked example

Solve dydx = xy given that y = 1 when x = 0.

1ydy= x dx separate: y on the left, x on the right ∫1ydy= ∫x dx integrate both sides, one constant is enough ln|y|= x²2 + c y= Ae^x²/2 A = e^c, which is why the constant becomes a multiplier 1= Ae⁰, so A = 1 and y = e^x²/2 at x = 2 that is e² ≈ 7.389

The constant does not stay where you put it. Exponentiating turns "+ c" into "× A". Students who carry a stray "+ c" through the exponential get an answer that does not satisfy the original equation, and the only way to notice is to check it.

The one that comes up most

dPdt= kP → P = P₀e^kt worth knowing on sight

With P₀ = 100 and k = 0.2, P(5) = 100e ≈ 271.8. Positive k grows, negative k decays, and the size of k sets how fast.

Your turn

1. "A culture doubles in size every hour" suggests which equation?

dN/dt = 2 dN/dt = kN dN/dt = 2t dN/dt = k/N

2. With P₀ = 100 and k = 0.2, find P(5) to 1 decimal place.

3. For dy/dx = xy with y(0) = 1, find y when x = 2, to 3 decimal places.

Where the marks go

Setting up the equation from the words is usually the first mark and it is where most of the difficulty is. Define your letters before you use them.

Separating correctly, with the integral signs written on both sides, is the method mark. Jumping straight to an exponential answer loses it even when the answer is right.

Use the initial condition. A general solution with an unfound constant is an incomplete answer whenever a condition was given.

Want a verdict on your own draft?

These pages are free and stay free, but they are general and your IA is not. Send me your research question, or whatever exists so far, and I will tell you in writing whether the topic has a ceiling on it, where the marks are going, and what to change first. That costs nothing and it comes back within 24 hours.

Written by a serving IB Diploma and Career-related Programme Coordinator and Head of Mathematics, who reads internal assessments across every subject group every year. If you then want the whole draft reviewed properly against all five criteria, that is the paid one, and it is refunded if it does not name at least three specific things to fix.

Get a free verdict Full written review, $99

I never write any part of it. Not a sentence, not a calculation, not your data. Under 18: a parent buys this and the thread is with them. I do not work with students at my own school.