Standard +0.3 This is a standard separable variables differential equation requiring routine integration techniques (partial fractions for the y-side, standard trig identity for the x-side), followed by finding stationary points by setting dy/dx = 0. While it involves multiple steps, each is a well-practiced A-level technique with no novel insight required, making it slightly easier than average.
10
\includegraphics[max width=\textwidth, alt={}, center]{3621a7e5-a3fb-42c1-828d-7068fddbf2f9-3_677_691_781_724}
A particular solution of the differential equation
$$3 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = 4 \left( y ^ { 3 } + 1 \right) \cos ^ { 2 } x$$
is such that \(y = 2\) when \(x = 0\). The diagram shows a sketch of the graph of this solution for \(0 \leqslant x \leqslant 2 \pi\); the graph has stationary points at \(A\) and \(B\). Find the \(y\)-coordinates of \(A\) and \(B\), giving each coordinate correct to 1 decimal place.
Separate variables and integrate at least one side
M1
Obtain \(\ln(y^3 + 1) = \ldots\) or equivalent
A1
Obtain \(\ldots = 2x + \sin 2x\) or equivalent
A1
Use \(x = 0, y = 2\) to find constant of integration (or as limits) in an expression containing at least two terms of the form \(\ln(y^3 + 1), bx\) or \(c \sin 2x\)
M1*
Obtain \(\ln(y^3 + 1) = 2x + \sin 2x + \ln 9\) or equivalent e.g. implied by correct constant
A1
Identify at least one of \(\frac{1}{2}\pi\) and \(\frac{3}{2}\pi\) as \(x\)-coordinate at stationary point
B1
Use correct process to find \(y\)-coordinate for at least one \(x\)-coordinate
M1(d*M)
Obtain \(5.9\)
A1
Obtain \(48.1\)
A1
[10]
Use $2\cos^2 x = 1 + \cos 2x$ or equivalent | B1 |
Separate variables and integrate at least one side | M1 |
Obtain $\ln(y^3 + 1) = \ldots$ or equivalent | A1 |
Obtain $\ldots = 2x + \sin 2x$ or equivalent | A1 |
Use $x = 0, y = 2$ to find constant of integration (or as limits) in an expression containing at least two terms of the form $\ln(y^3 + 1), bx$ or $c \sin 2x$ | M1* |
Obtain $\ln(y^3 + 1) = 2x + \sin 2x + \ln 9$ or equivalent e.g. implied by correct constant | A1 |
Identify at least one of $\frac{1}{2}\pi$ and $\frac{3}{2}\pi$ as $x$-coordinate at stationary point | B1 |
Use correct process to find $y$-coordinate for at least one $x$-coordinate | M1(d*M) |
Obtain $5.9$ | A1 |
Obtain $48.1$ | A1 | [10]
10\\
\includegraphics[max width=\textwidth, alt={}, center]{3621a7e5-a3fb-42c1-828d-7068fddbf2f9-3_677_691_781_724}
A particular solution of the differential equation
$$3 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = 4 \left( y ^ { 3 } + 1 \right) \cos ^ { 2 } x$$
is such that $y = 2$ when $x = 0$. The diagram shows a sketch of the graph of this solution for $0 \leqslant x \leqslant 2 \pi$; the graph has stationary points at $A$ and $B$. Find the $y$-coordinates of $A$ and $B$, giving each coordinate correct to 1 decimal place.
\hfill \mbox{\textit{CAIE P3 2013 Q10 [10]}}