2026 RI P2 Q4

Do not use a calculator in answering this question.

1. The curve \(y=\mathrm{f}(x)\) passes through the origin and has gradient given by
   \[\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{x+y}{1+xy}.\]
   1. Show that \(\displaystyle(1+xy)\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}+x\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)^2+(y-1)\dfrac{\mathrm{d}y}{\mathrm{d}x}=1\).[2]
   2. By further differentiation of the result in part (a)(i), show that
      \[(1+xy)\frac{\mathrm{d}^3y}{\mathrm{d}x^3}+\mathrm{p}(x)\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)\left(\frac{\mathrm{d}^2y}{\mathrm{d}x^2}\right)+\mathrm{q}(y)\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+r\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2=0,\]
      where the functions \(\mathrm{p}(x)\), \(\mathrm{q}(y)\) and the constant \(r\) are to be determined.[3]
   3. Hence find the Maclaurin series for \(y\) up to and including the term in \(x^3\).[2]
2. The equation
   \[2\sqrt{2}\cos\left(x-\frac\pi4\right)+3\sin2x-3=0\]
   has a root \(\theta\), which is close to zero.
   1. Show that \(\theta^2+a\theta+b\approx0\), where \(a\) and \(b\) are integers to be determined.[3]
   2. Hence find an approximation for \(\theta\) in surd form.[2]