Series solutions of differential equationsEdexcel International A Level Further Maths: Subtopic test

1. 1
   The function $y(x)$ satisfies the differential equation $\frac{d^2y}{dx^2}=x+2y$, with $y=1$ and $\frac{dy}{dx}=2$ at $x=0$. A series solution in ascending powers of $x$ is to be found.

   (a)

   Find the value of $\frac{d^3y}{dx^3}$ at $x=0$.

   [1 mark]

   • A$5$
   • B$4$
   • C$2$
   • D$1$

   (b)

   Find the value of $\frac{d^4y}{dx^4}$ at $x=0$.

   [1 mark]

   • A$5$
   • B$2$
   • C$4$
   • D$8$

   (c)

   Find the series solution up to and including the term in $x^4$, and use it to estimate $y(0.2)$ to 4 decimal places.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $y(x)$ satisfies the differential equation $\frac{d^2y}{dx^2}=xy$, with $y=1$ and $\frac{dy}{dx}=1$ at $x=0$.

   (a)

   Which expression is equal to $\frac{d^3y}{dx^3}$?

   [1 mark]

   • A$y'+xy$
   • B$y+xy'$
   • C$xy''$
   • D$2y+xy'$

   (b)

   Find the value of $\frac{d^4y}{dx^4}$ at $x=0$.

   [1 mark]

   • A$1$
   • B$4$
   • C$0$
   • D$2$

   (c)

   Find the series solution up to and including the term in $x^4$, and use it to estimate $y(0.4)$ to 3 decimal places.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $y(x)$ satisfies the first-order differential equation $\frac{dy}{dx}=x^2+y$, with $y=1$ at $x=0$.

   (a)

   Find the values of $\frac{dy}{dx}$, $\frac{d^2y}{dx^2}$ and $\frac{d^3y}{dx^3}$ at $x=0$.

   [3 marks]

   (b)

   Find the series solution for $y$ in ascending powers of $x$ up to and including the term in $x^4$, and use it to estimate $y(0.2)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $y(x)$ satisfies the differential equation $\frac{d^2y}{dx^2}=x\frac{dy}{dx}+y^2$, with $y=1$ and $\frac{dy}{dx}=1$ at $x=0$.

   (a)

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

   [6 marks]

   (b)

   (i) Differentiate your series to find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$, and verify that $\frac{d^2y}{dx^2}-x\frac{dy}{dx}-y^2$ has no terms in $x^0$, $x^1$ or $x^2$.
   (ii) Use your series to estimate $y(0.1)$ to 4 decimal places.

   [6 marks]

   Total for question 4: 12 marks

End of questions