Further Maths differential equations test to download as PDF

This Further Maths differential equations test lasts 60 minutes and is marked out of 20. It covers the whole chapter for pupils aged 16-18, and the full answers are on a separate page.

⏱️ Duration : 60 min
🎯 Total : 20 marks
❓ 5 questions

Mark/20

Question 1 – First order equation
Solve \(\dfrac{dy}{dx}-\dfrac{3y}x=x^2\) for \(x>0\), given that \(y=2\) when \(x=1\). [4]
Question 2 – Second order with initial conditions
(a) Find the general solution of \(y''-5y'+6y=4e^{x}\). [4]
(b) Given that \(y=5\) and \(y'=7\) when \(x=0\), find the particular solution. [2]
Question 3 – Complex roots
Solve \(\dfrac{d^2y}{dx^2}+4\dfrac{dy}{dx}+20y=0\), given that \(y=2\) and \(\dfrac{dy}{dx}=0\) when \(x=0\). [4]
Question 4 – Simple harmonic motion
A particle moves with \(\ddot x=-25x\) and amplitude 0.2 m.
1. Find the period. [1]
2. Find the greatest speed. [1]
3. Find the speed when \(x=0.12\) m. [1]
Question 5 – Problem: the cooling engine
A hot engine at \(75^\circ\)C cools in a garage at \(15^\circ\)C so that \(\dfrac{dT}{dt}=-0.05(T-15)\), where \(t\) is in minutes.
1. Show that \(T=15+60e^{-0.05t}\). [2]
2. Find how long the engine takes to cool to \(30^\circ\)C. [1]

How to use this Further Maths differential equations test

First, revise with the differential equations lesson and the worksheet. Then sit the test in one go, with a timer set to 60 minutes. After that, mark your work with the differential equations test answers. If you lose marks on a question, redo it the next day without help.

Finally, you can download this Further Maths differential equations test as a PDF and print it like a real exam paper. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday.