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<p>The variables \(x\) and \(y\) satisfy the differential equation \(\frac{\mathrm{d}y}{\mathrm{d}x}=y^3-\frac{y}{x}\), where&nbsp;\(x,y&gt;0\). It is given that&nbsp;\(y=2\) when&nbsp;\(x=1\).</p>
<ol class="abc-list"><li data-list-item-id="p2-tzc-q12-item01" data-list-item-path="p2-tzc-q12-item01"><div class="list-item-paragraph">Use Euler’s method with step size \(0.01\) to find an approximation for&nbsp;\(y\) when&nbsp;\(x=1.05\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p2-tzc-q12-item02" data-list-item-path="p2-tzc-q12-item02"><div class="list-item-paragraph">The substitution \(\frac1{y^2}=-2z\) is used.</div><div class="list-item-paragraph">Show that&nbsp;\(\frac{\mathrm{d}z}{\mathrm{d}x}-\frac{2z}{x}=1\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="p2-tzc-q12-item03" data-list-item-path="p2-tzc-q12-item03"><div class="list-item-paragraph">Hence solve the differential equation, giving \(y\) as a function of \(x\).<span class="inline-marks">[7]</span></div></li><li data-list-item-id="p2-tzc-q12-item04" data-list-item-path="p2-tzc-q12-item04"><div class="list-item-paragraph">State the domain and range of the solution.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p2-tzc-q12-item05" data-list-item-path="p2-tzc-q12-item05"><div class="list-item-paragraph"><p>By considering the shape of the graph of the solution, state whether the approximation in part <strong>(a)</strong> is an overestimate or an underestimate, giving a reason.</p><span class="inline-marks">[1]</span></div></li></ol>

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 \(x=1\).

Use Euler’s method with step size \(0.01\) to find an approximation for \(y\) when \(x=1.05\).

Hence solve the differential equation, giving \(y\) as a function of \(x\).

State the domain and range of the solution.

Euler's Method

Function Notation, Domain and Range

Solving Differential Equations by Substitution

Concavity of Graphs

Linear First-Order Differential Equations and Integrating Factors

IB AA HL May 2026 P2 TZC Q12

Functions

Differential Equations

Applications of Differentiation

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