1. 1
   The function $y=f(x)$ satisfies the differential equation $\dfrac{dy}{dx}=x+y$, with $y=1$ when $x=0$. Euler's method with step length $h=0.1$ is used to approximate $f(0.2)$, with $x_0=0$ and $y_0=1$.

   (a)

   Find $y_1$.

   [1 mark]

   • A$1.1$
   • B$1.11$
   • C$0.1$
   • D$2$

   (b)

   Find $y_2$, the Euler approximation to $f(0.2)$.

   [1 mark]

   • A$1.21$
   • B$1.2$
   • C$0.12$
   • D$1.22$

   (c)

   By considering $\dfrac{d^{2}y}{dx^{2}}$, determine whether $y_2$ is an underestimate or an overestimate of $f(0.2)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the differential equation $\dfrac{dy}{dx}=3x^{2}y$.

   (a)

   Which of the following is the general solution of the differential equation, where $A$ is an arbitrary constant?

   [1 mark]

   • A$y=e^{x^{3}}+A$
   • B$y=x^{3}y+A$
   • C$y=Ae^{x^{3}}$
   • D$y=Ae^{3x^{2}}$

   (b)

   Given that $y=2$ when $x=0$, find the value of $y$ when $x=1$.

   [1 mark]

   • A$2e$
   • B$e+1$
   • C$e$
   • D$e^{3}$

   (c)

   For the solution with $y=2$ when $x=0$, find the exact value of $x$ for which $y=10$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the differential equation $\dfrac{dy}{dx}=\dfrac{x^{2}+y^{2}}{xy}$, for $x>0$ and $y>0$, with $y=2$ when $x=1$.

   (a)

   Use the substitution $y=vx$ to show that $x\dfrac{dv}{dx}=\dfrac{1}{v}$.

   [3 marks]

   (b)

   Hence find $y$ in terms of $x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A gardener models the height, $h$ cm, of a potted herb $t$ weeks after it is planted out. When $t=0$, $h=5$. Two models are proposed.
   Model 1: $\dfrac{dh}{dt}=0.02h(50-h)$.
   Model 2: $\dfrac{dh}{dt}+\dfrac{h}{t+1}=6$.

   (a)

   For Model 1, show that $h=\dfrac{50}{1+9e^{-t}}$.

   [6 marks]

   (b)

   (i) For Model 2, use an integrating factor to find $h$ in terms of $t$.
   (ii) Compare the long-term behaviour of the two models and state, with a reason, which model is more realistic.

   [6 marks]

   Total for question 4: 12 marks

End of questions