Topic 10: Differential equations

Question commentary

Question	Inspired by	Contents
Qn 1001	-	General and particular solutions of differential equations of the forms: $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}=f(x)$ $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}=f(y)$ $\displaystyle \frac{\mathrm{d}^2y}{\mathrm{d}x^2}=f(x)$
Qn 1002	-	Problem sum: formulating a differential equation from a problem situation differential equation of the form $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}=f(y)$ interpreting a differential equation and its solution in terms of a problem situation
Qn 1003	-	The substitution technique: reducing a differential equation to the following forms by means of a given substitution $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}=f(x)$ $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}=f(y)$

We believe our suite of 3 questions (with at least 39 unique question types) is relatively complete in coverage when measuring against the syllabus document as well as the past TYS questions.

That said, the most common difficulty students face is the ability to translate problem situations described in words into the mathematical symbols of a differential equation. We believe that improving in this aspect takes exposure to different problem sums. Computer generated questions may not be the best approach here to achieve competency in this aspect (as opposed to the more mechanical process of solving a DE that has already been formulated).

TYS analysis

Year	Question	Math Atlas	Comments
2007	P1 Q4	Qn 1002	Problem sum with logarithmic integration
2008	P1 Q4	Qn 1001	$\frac{f'(x)}{f(x)}$ integration
2009	P2 Q4	Qn 1002 Qn 1002	Second order DE, problem sum with logarithmic integration
2010	P1 Q7	Qn 1002	Problem sum with logarithmic integration
2011	P1 Q8	Qn 1001	Problem sum, integration using MF26 formula
2012	P2 Q1	Qn 1001	Second order DE and first order DE integrated using MF26 formula
2013	P1 Q10	Qn 1003	Extension of the substitution technique
2014	P1 Q10	Qn 1001	Integration using completing the square and MF26 formula
2015	P2 Q1	Qn 1001	Problem sum with integration of square roots
2016	P1 Q9	Qn 1001 Qn 1003	Problem sum with the substitution technique, second order DE
2017	Specimen P1 Q11	Qn 1002	Problem sum with completing the square/partial fraction integration
2017	P1 Q11	Qn 1002	Problem sum with logarithmic integration
2018	P2 Q1	Qn 1001	Use of the substitution technique
2018	P2 Q1	Qn 1001	Integration of cube roots
2019	P1 Q11	Qn 1002	Problem sum with logarithmic integration
2020	P1 Q11	Qn 1002	Problem sum with logarithmic integration