Further Pure 1: Further differential equationsEdexcel A-Level Further Maths: Topic test

1. 1
   The function $y$ satisfies $\frac{dy}{dx}=x+y$, with $y=1$ when $x=0$.

   (a)

   Find the value of $\frac{d^2y}{dx^2}$ when $x=0$.

   [1 mark]

   • A$1$
   • B$3$
   • C$4$
   • D$2$

   (b)

   What is the coefficient of $x^3$ in the series solution?

   [1 mark]

   • A$2$
   • B$\frac13$
   • C$\frac23$
   • D$\frac16$

   (c)

   Find the term in $x^4$ in the series solution.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The differential equation $y\frac{dy}{dx}+y^2=\mathrm{e}^{-x}$ is to be solved for $y>0$ using the substitution $z=y^2$.

   (a)

   Which equation does $z$ satisfy?

   [1 mark]

   • A$\frac{dz}{dx}+2z=2\mathrm{e}^{-x}$
   • B$\frac{dz}{dx}+z=\mathrm{e}^{-x}$
   • C$\frac{dz}{dx}+2z=\mathrm{e}^{-x}$
   • D$\frac{dz}{dx}+2\sqrt{z}=2\mathrm{e}^{-x}$

   (b)

   What is the integrating factor for the equation in $z$?

   [1 mark]

   • A$\mathrm{e}^{-2x}$
   • B$\mathrm{e}^{x}$
   • C$\mathrm{e}^{2x}$
   • D$\mathrm{e}^{2z}$

   (c)

   Given that $y=2$ when $x=0$, find $y^2$ in terms of $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $y$ satisfies $\frac{d^2y}{dx^2}=xy$, with $y=1$ and $\frac{dy}{dx}=1$ when $x=0$.

   (a)

   Show that $\frac{d^3y}{dx^3}=y+x\frac{dy}{dx}$ and find a similar expression for $\frac{d^4y}{dx^4}$.

   [3 marks]

   (b)

   Find the series solution for $y$ in ascending powers of $x$, up to and including the term in $x^4$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Consider the differential equation $\frac{d^2y}{dx^2}+\frac{2}{x}\frac{dy}{dx}+y=0$ for $x>0$.

   (a)

   Show that the substitution $y=\frac{z}{x}$ transforms the equation into $\frac{d^2z}{dx^2}+z=0$, and hence find the general solution for $y$.

   [6 marks]

   (b)

   (i) Given that $y=-\frac{1}{\pi}$ and $\frac{dy}{dx}=\frac{1-2\pi}{\pi^2}$ when $x=\pi$, find $y$ in terms of $x$. [4]
   (ii) Describe the behaviour of $y$ as $x\to\infty$. [2]

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The function $y$ satisfies $\frac{dy}{dx}=\mathrm{e}^x-2y$, with $y=1$ when $x=0$.

   (a)

   Which expression is equal to $\frac{d^2y}{dx^2}$?

   [1 mark]

   • A$\mathrm{e}^x-2y$
   • B$\mathrm{e}^x-2\frac{dy}{dx}$
   • C$\mathrm{e}^x+2\frac{dy}{dx}$
   • D$\mathrm{e}^x-2$

   (b)

   What is the value of $\frac{d^3y}{dx^3}$ when $x=0$?

   [1 mark]

   • A$5$
   • B$-1$
   • C$11$
   • D$-5$

   (c)

   Find the series solution for $y$ in ascending powers of $x$, up to and including the term in $x^3$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Consider the differential equation $x^2\frac{d^2y}{dx^2}+3x\frac{dy}{dx}+y=0$ for $x>0$, using the substitution $x=\mathrm{e}^t$.

   (a)

   Which equation does $y$ satisfy as a function of $t$?

   [1 mark]

   • A$\frac{d^2y}{dt^2}+3\frac{dy}{dt}+y=0$
   • B$\frac{d^2y}{dt^2}+4\frac{dy}{dt}+y=0$
   • C$\frac{d^2y}{dt^2}+2\frac{dy}{dt}+y=0$
   • D$\frac{d^2y}{dt^2}-2\frac{dy}{dt}+y=0$

   (b)

   Which expression is the general solution in terms of $x$?

   [1 mark]

   • A$y=\dfrac{A+B\ln x}{x}$
   • B$y=(A+B\ln x)x$
   • C$y=\dfrac{A+Bx}{x}$
   • D$y=Ax+B\ln x$

   (c)

   Given that $y=3$ and $\frac{dy}{dx}=-2$ when $x=1$, find $y$ in terms of $x$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The function $y$ satisfies $\frac{d^2y}{dx^2}=2y\frac{dy}{dx}$, with $y=1$ and $\frac{dy}{dx}=1$ when $x=0$.

   (a)

   Show that $\frac{d^3y}{dx^3}=2\left(\frac{dy}{dx}\right)^2+2y\frac{d^2y}{dx^2}$, and find an expression for $\frac{d^4y}{dx^4}$.

   [3 marks]

   (b)

   Find the series solution for $y$ in ascending powers of $x$, up to and including the term in $x^4$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The function $y$ satisfies $\frac{dy}{dx}+y=y^2\mathrm{e}^x$, with $y=1$ when $x=0$.

   (a)

   Use the Taylor series method to find the series solution for $y$ in ascending powers of $x$, up to and including the term in $x^3$.

   [6 marks]

   (b)

   Use the substitution $z=\frac1y$ to find $y$ in terms of $x$, and verify that your solution agrees with the series in part (a) up to the term in $x^3$.

   [6 marks]

   Total for question 8: 12 marks

End of questions