FP2: Second order differential equationsEdexcel International A Level Further Maths: Topic test

1. 1
   The function $y(x)$ satisfies the differential equation $\frac{d^2y}{dx^2}+6\frac{dy}{dx}+13y=0$.

   (a)

   Find the roots of the auxiliary equation.

   [1 mark]

   • A$-3\pm4\mathrm{i}$
   • B$3\pm2\mathrm{i}$
   • C$-3\pm2\mathrm{i}$
   • D$-6\pm4\mathrm{i}$

   (b)

   Which expression is the general solution of the differential equation?

   [1 mark]

   • A$y=\mathrm{e}^{-3x}(A\cos2x+B\sin2x)$
   • B$y=\mathrm{e}^{3x}(A\cos2x+B\sin2x)$
   • C$y=\mathrm{e}^{-3x}(A\cos4x+B\sin4x)$
   • D$y=\mathrm{e}^{-2x}(A\cos3x+B\sin3x)$

   (c)

   Given that $y=0$ and $\frac{dy}{dx}=4$ when $x=0$, find $y$ in terms of $x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The differential equation $\frac{d^2y}{dx^2}+4\frac{dy}{dx}+5y=8\mathrm{e}^{-2x}$ is to be solved using the substitution $y=z\mathrm{e}^{-2x}$, where $z$ is a function of $x$.

   (a)

   Find $\frac{dy}{dx}$ in terms of $z$ and $\frac{dz}{dx}$.

   [1 mark]

   • A$\left(\frac{dz}{dx}+2z\right)\mathrm{e}^{-2x}$
   • B$\frac{dz}{dx}\mathrm{e}^{-2x}$
   • C$-2\frac{dz}{dx}\mathrm{e}^{-2x}$
   • D$\left(\frac{dz}{dx}-2z\right)\mathrm{e}^{-2x}$

   (b)

   The substitution transforms the differential equation into which equation?

   [1 mark]

   • A$\frac{d^2z}{dx^2}+z=8\mathrm{e}^{-2x}$
   • B$\frac{d^2z}{dx^2}+z=8$
   • C$\frac{d^2z}{dx^2}+4\frac{dz}{dx}+z=8$
   • D$\frac{d^2z}{dx^2}+9z=8$

   (c)

   Solve the transformed equation and hence find the general solution for $y$ in terms of $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The differential equation $\frac{d^2y}{dx^2}-2\frac{dy}{dx}+10y=10x-3$.

   (a)

   Find the complementary function.

   [3 marks]

   (b)

   Hence find the general solution of the differential equation.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $y(x)$, defined for $x>0$, satisfies the differential equation $x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}-4y=3x^2$.

   (a)

   Use the substitution $x=\mathrm{e}^t$ to show that the differential equation can be written as $\frac{d^2y}{dt^2}-4y=3\mathrm{e}^{2t}$, and find the general solution of this equation for $y$ in terms of $t$.

   [6 marks]

   (b)

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

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The differential equation $x^2\frac{d^2y}{dx^2}-x\frac{dy}{dx}+5y=0$ is to be solved for $x>0$ using the substitution $x=\mathrm{e}^t$.

   (a)

   Express $x^2\frac{d^2y}{dx^2}$ in terms of derivatives of $y$ with respect to $t$.

   [1 mark]

   • A$\frac{d^2y}{dt^2}$
   • B$\frac{d^2y}{dt^2}-\frac{dy}{dt}$
   • C$\frac{d^2y}{dt^2}+\frac{dy}{dt}$
   • D$\mathrm{e}^{2t}\frac{d^2y}{dt^2}$

   (b)

   The transformed equation is a constant-coefficient equation in $y$ and $t$. What is its auxiliary equation?

   [1 mark]

   • A$m^2-m+5=0$
   • B$m^2+2m+5=0$
   • C$m^2-2m-5=0$
   • D$m^2-2m+5=0$

   (c)

   Hence find the general solution for $y$ in terms of $x$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The function $y(x)$ satisfies the differential equation $\frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=20\mathrm{e}^{2x}$.

   (a)

   Which expression is the complementary function?

   [1 mark]

   • A$y=A\mathrm{e}^{2x}+B\mathrm{e}^{-3x}$
   • B$y=A\mathrm{e}^{-2x}+B\mathrm{e}^{3x}$
   • C$y=A\mathrm{e}^{x}+B\mathrm{e}^{-6x}$
   • D$y=(A+Bx)\mathrm{e}^{2x}$

   (b)

   Which is the correct form to try for a particular integral?

   [1 mark]

   • A$k\mathrm{e}^{2x}$
   • B$kx^2\mathrm{e}^{2x}$
   • C$kx\mathrm{e}^{2x}$
   • D$k\mathrm{e}^{20x}$

   (c)

   Find the particular integral, using the form from part (b).

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The differential equation $\frac{d^2y}{dx^2}+4\frac{dy}{dx}+3y=10\sin x$.

   (a)

   Find a particular integral.

   [3 marks]

   (b)

   Given that $y=0$ and $\frac{dy}{dx}=1$ when $x=0$, find $y$ in terms of $x$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A particle moves in a straight line. Its displacement $x$ metres from a fixed point at time $t$ seconds satisfies $\frac{d^2x}{dt^2}+2\frac{dx}{dt}+5x=10$, with $x=0$ and $\frac{dx}{dt}=0$ when $t=0$.

   (a)

   Find $x$ in terms of $t$.

   [6 marks]

   (b)

   Show that $\frac{dx}{dt}=5\mathrm{e}^{-t}\sin2t$, and hence find the greatest displacement of the particle, giving your answer to 3 significant figures. Justify that this is the greatest value.

   [6 marks]

   Total for question 8: 12 marks

End of questions