First order equations and integrating factorsAQA A-Level Further Maths: Subtopic test

1. 1
   $\frac{dy}{dx}+\frac{3}{x}y=x^2$, for $x>0$.

   (a)

   Find an integrating factor for this differential equation.

   [1 mark]

   • A$e^{3x}$
   • B$3\ln x$
   • C$x^3$
   • D$x^{-3}$

   (b)

   After multiplying by the integrating factor, which equation can be integrated directly?

   [1 mark]

   • A$\frac{d}{dx}(x^3y)=x^5$
   • B$\frac{d}{dx}(x^3y)=x^2$
   • C$\frac{d}{dx}\left(\frac{y}{x^3}\right)=x^2$
   • D$\frac{d}{dx}(x^3y)=x^3$

   (c)

   Find the general solution, giving $y$ in terms of $x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $\frac{dy}{dx}-2y=e^{3x}$.

   (a)

   Find an integrating factor for this differential equation.

   [1 mark]

   • A$e^{2x}$
   • B$-2x$
   • C$e^{3x}$
   • D$e^{-2x}$

   (b)

   Find the general solution.

   [1 mark]

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

   (c)

   Given that $y=3$ when $x=0$, find $y$ in terms of $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   $\frac{dy}{dx}+y\tan x=\cos x$, for $-\frac{\pi}{2}<x<\frac{\pi}{2}$.

   (a)

   Show that the general solution is $y=(x+c)\cos x$.

   [3 marks]

   (b)

   Given that $y=3$ when $x=0$, find the exact value of $y$ when $x=\frac{\pi}{3}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle moves in a straight line. At time $t$ seconds, $t\geq0$, its velocity is $v$ m s$^{-1}$, where $\frac{dv}{dt}+2v=10e^{-t}$ and $v=0$ when $t=0$.

   (a)

   (i) Find $v$ in terms of $t$.
   (ii) Find the maximum velocity of the particle.

   [6 marks]

   (b)

   The particle starts at the point $O$. Find its displacement $s$ metres from $O$ at time $t$, and use it to show that the particle never travels more than 5 m from $O$.

   [6 marks]

   Total for question 4: 12 marks

End of questions