Standard +0.3 This is a straightforward separable variables question requiring separation, integration using standard identities (cos²3y = (1+cos6y)/2), and applying initial conditions. While it involves multiple steps and the double angle formula, it follows a completely standard template with no novel insight required, making it slightly easier than average.
8 The variables \(x\) and \(y\) satisfy the differential equation
$$\mathrm { e } ^ { 4 x } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \cos ^ { 2 } 3 y .$$
It is given that \(y = 0\) when \(x = 2\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
Separate variables correctly and reach \(a\sec^2 3y\) or \(be^{-4x}\)
B1
Condone missing integral signs or \(dy\) and \(dx\), but allow if recognisable integrals follow. Not for \(1/\cos^2 3y\) and \(1/e^{4x}\).
Obtain term \(-\frac{1}{4}e^{-4x}\)
B1
Can recover the previous B1 if \(de^{-4x}\) seen here.
Obtain only a term of the form \(a\tan 3y\)
M1
Can recover the first B1 if \(a\tan 3y\) seen here.
Obtain term \(\frac{1}{3}\tan 3y\)
A1
Use \(x=2\), \(y=0\) to evaluate a constant or as limits in a solution containing terms of the form \(a\tan by\) and \(ce^{\pm 4x}\)
M1
May see \(\tan by\) and \(e^{\pm 4x}\) here.
Obtain correct answer in any form
A1
e.g. \(\frac{1}{3}\tan 3y = -\frac{1}{4}e^{-4x}+\frac{1}{4}e^{-8}\) or \(\frac{1}{3}\tan 3y = -\frac{1}{4}e^{-4x}+8.39\times 10^{-5}\)
Obtain final answer \(y=\frac{1}{3}\tan^{-1}\left(\frac{3}{4}e^{-8}-\frac{3}{4}e^{-4x}\right)\)
A1
ISW. OE e.g. \(y=\frac{1}{3}\tan^{-1}\left(2.52\times10^{-4}-\frac{3}{4}e^{-4x}\right)\)
Total
7
## Question 8:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Separate variables correctly and reach $a\sec^2 3y$ or $be^{-4x}$ | B1 | Condone missing integral signs or $dy$ and $dx$, but allow if recognisable integrals follow. Not for $1/\cos^2 3y$ and $1/e^{4x}$. |
| Obtain term $-\frac{1}{4}e^{-4x}$ | B1 | Can recover the previous B1 if $de^{-4x}$ seen here. |
| Obtain only a term of the form $a\tan 3y$ | M1 | Can recover the first B1 if $a\tan 3y$ seen here. |
| Obtain term $\frac{1}{3}\tan 3y$ | A1 | |
| Use $x=2$, $y=0$ to evaluate a constant or as limits in a solution containing terms of the form $a\tan by$ and $ce^{\pm 4x}$ | M1 | May see $\tan by$ and $e^{\pm 4x}$ here. |
| Obtain correct answer in any form | A1 | e.g. $\frac{1}{3}\tan 3y = -\frac{1}{4}e^{-4x}+\frac{1}{4}e^{-8}$ or $\frac{1}{3}\tan 3y = -\frac{1}{4}e^{-4x}+8.39\times 10^{-5}$ |
| Obtain final answer $y=\frac{1}{3}\tan^{-1}\left(\frac{3}{4}e^{-8}-\frac{3}{4}e^{-4x}\right)$ | A1 | ISW. OE e.g. $y=\frac{1}{3}\tan^{-1}\left(2.52\times10^{-4}-\frac{3}{4}e^{-4x}\right)$ |
| **Total** | **7** | |
8 The variables $x$ and $y$ satisfy the differential equation
$$\mathrm { e } ^ { 4 x } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \cos ^ { 2 } 3 y .$$
It is given that $y = 0$ when $x = 2$.\\
Solve the differential equation, obtaining an expression for $y$ in terms of $x$.\\
\hfill \mbox{\textit{CAIE P3 2023 Q8 [7]}}