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Differential equations — first-order separable and Euler's method (IB Math AI HL)
Updated October 2026
Quick answer: A separable equation dy/dx = f(x)g(y) is solved by separating the variables, (1/g(y)) dy = f(x) dx, and integrating both sides with one constant. When it cannot be separated, Euler's method steps along the curve: yₙ₊₁ = yₙ + h·f(xₙ, yₙ) and xₙ₊₁ = xₙ + h. A smaller step h usually gives a better estimate.
HL AI's Paper 3 favourite. Separable DEs by algebra, non-separable by Euler's numerical method. Practise the CG50 keystrokes for Euler.
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5 quick questions on separable differential equations. No sign-in needed.
\(\frac{dy}{dx} = 2xy\) and \(y = 3\) when \(x = 0\). Find \(y\) when \(x = 1\).
Starting from the point \((1,\ 4)\) on a solution of \(\frac{dy}{dx} = 3x - y\), take two Euler steps of size \(0.2\). What estimate of \(y(1.4)\) do you get?
\(\frac{dy}{dx} = \frac{x}{y}\), \(y > 0\), and \(y = 3\) when \(x = 0\). Find \(y\) when \(x = 4\).
Starting from the point \((2,\ 1)\) on a solution of \(\frac{dy}{dx} = x^2 - y\), take two Euler steps of size \(0.1\). What estimate of \(y(2.2)\) do you get?
\(\frac{dy}{dx} = 2y\) and \(y = 5\) when \(x = 0\). Find \(y\) when \(x = 3\).
Where can I practise differential equations questions?
On the Advanced Differentiation and Related Rates questions page: 40 exam-style HL AI questions with full mark schemes, plus free step-by-step worked examples.
Can I use my calculator for differential equations?
Yes: a GDC is allowed on every HL AI paper, so practise the calculator method as well as the working.