Standard +0.3 This is a separable differential equation requiring standard techniques: separation of variables, integration using trigonometric identities (cos²y = ½(1+cos2y) leads to sec²y form, tan x integrates to -ln|cos x|), and applying initial conditions. While it involves multiple steps and careful algebraic manipulation, it follows a well-practiced procedure with no novel insight required, making it slightly easier than average.
The variables \(x\) and \(y\) satisfy the differential equation
$$\frac{dy}{dx} = 4 \cos^2 y \tan x,$$
for \(0 \leqslant x < \frac{1}{2}\pi\), and \(x = 0\) when \(y = \frac{1}{4}\pi\). Solve this differential equation and find the value of \(x\) when \(y = \frac{1}{8}\pi\). [8]
Separate variables correctly and attempt integration of one side
B1
Obtain term tany, or equivalent
B1
Obtain term of the form klncosx, or equivalent
M1
Obtain term −4lncosx, or equivalent
A1
Use x = 0 and y= 1π in solution containing atany and blncosxto evaluate a
4
Answer
Marks
constant, or as limits
M1
Obtain correct solution in any form, e.g.tany=4lnsecx+1
A1
Substitutey=1πin solution containing terms atany and blncosx, and use correct
3
Answer
Marks
method to find x
M1
Obtain answer x = 0.587
A1
8
Answer
Marks
Guidance
Question
Answer
Marks
Question 6:
6 | Separate variables correctly and attempt integration of one side | B1
Obtain term tany, or equivalent | B1
Obtain term of the form klncosx, or equivalent | M1
Obtain term −4lncosx, or equivalent | A1
Use x = 0 and y= 1π in solution containing atany and blncosxto evaluate a
4
constant, or as limits | M1
Obtain correct solution in any form, e.g.tany=4lnsecx+1 | A1
Substitutey=1πin solution containing terms atany and blncosx, and use correct
3
method to find x | M1
Obtain answer x = 0.587 | A1
8
Question | Answer | Marks
The variables $x$ and $y$ satisfy the differential equation
$$\frac{dy}{dx} = 4 \cos^2 y \tan x,$$
for $0 \leqslant x < \frac{1}{2}\pi$, and $x = 0$ when $y = \frac{1}{4}\pi$. Solve this differential equation and find the value of $x$ when $y = \frac{1}{8}\pi$. [8]
\hfill \mbox{\textit{CAIE P3 2017 Q6 [8]}}