IB AA HL May 2026 P2 TZC Q12

The variables \(x\) and \(y\) satisfy the differential equation \(\frac{\mathrm{d}y}{\mathrm{d}x}=y^3-\frac{y}{x}\), where \(x,y>0\). It is given that \(y=2\) when \(x=1\).

1. Use Euler’s method with step size \(0.01\) to find an approximation for \(y\) when \(x=1.05\).[3]
2. The substitution \(\frac1{y^2}=-2z\) is used.
   Show that \(\frac{\mathrm{d}z}{\mathrm{d}x}-\frac{2z}{x}=1\).[4]
3. Hence solve the differential equation, giving \(y\) as a function of \(x\).[7]
4. State the domain and range of the solution.[3]

5. By considering the shape of the graph of the solution, state whether the approximation in part (a) is an overestimate or an underestimate, giving a reason.

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