Standard +0.3 This is a standard integrating factor question with straightforward identification of P(x) = -tan x and Q(x) = 2sec³x. The integrating factor is sec x (a common result), and while the integration of sec⁴x requires partial knowledge or recognition, this is a textbook application of the method with no novel problem-solving required. Slightly above average difficulty due to the trigonometric integration involved.
6.
$$\frac { \mathrm { d } y } { \mathrm {~d} x } - y \tan x = 2 \sec ^ { 3 } x$$
Given that \(y = 3\) at \(x = 0\), find \(y\) in terms of \(x\)
(Total 7 marks)
\(\left(\cos x\frac{dy}{dx} - y\sin x = 2\sec^2 x\right)\)
\(y\cos x = \int 2\sec^2 x dx\) (or equiv.) Or: \(\frac{d}{dx}(y\cos x) = 2\sec^2 x\)
M1A1(ft)
\(y\cos x = 2\tan x (+C)\) (or equiv.)
A1
\(y = 32\tan x + C = 3\)
M1
\(y = \frac{\cos x}{\text{(Or equiv. in the form }} y = f(x))\)
A17
1st M: Also scored for \(e^{\int \tan xdx} = e^{-\ln(\cos x)}\) (or \(e^{\ln\sec x}\)), then A0 for sec x
2nd M: Attempt to use their integrating factor (requires one side of the equation 'correct' for their integrating factor)
2nd A: The follow-through is allowed only in the case where the integrating factor used is \(\sec x\) or \(-\sec x\) \(\left(\int \sec x = \int 2\sec^4 xdx\right)\)
3rd M: Using \(y = 3\) at \(x = 0\) to find a value for \(C\) (dependent on an integration attempt, however poor, on the RHS)
Alternative:
Answer
Marks
1st M: Multiply through the given equation by \(\cos x\)
1st A: Achieving \(\cos x\frac{dy}{dx} - y\sin x = 2\sec^2 x\) (Allowing the possibility of integrating by inspection)
Total: [7]
Integrating factor: $e^{\int \tan xdx} = e^{\ln(\cos x)}$ (or $e^{-\ln\sec x}$), $= \cos x$ (or $\frac{1}{\sec x}$) | M1, A1 |
$\left(\cos x\frac{dy}{dx} - y\sin x = 2\sec^2 x\right)$ | |
$y\cos x = \int 2\sec^2 x dx$ (or equiv.) Or: $\frac{d}{dx}(y\cos x) = 2\sec^2 x$ | M1A1(ft) |
$y\cos x = 2\tan x (+C)$ (or equiv.) | A1 |
$y = 32\tan x + C = 3$ | M1 |
$y = \frac{\cos x}{\text{(Or equiv. in the form }} y = f(x))$ | A17 |
**1st M:** Also scored for $e^{\int \tan xdx} = e^{-\ln(\cos x)}$ (or $e^{\ln\sec x}$), then A0 for sec x | |
**2nd M:** Attempt to use their integrating factor (requires one side of the equation 'correct' for their integrating factor) | |
**2nd A:** The follow-through is allowed only in the case where the integrating factor used is $\sec x$ or $-\sec x$ $\left(\int \sec x = \int 2\sec^4 xdx\right)$ | |
**3rd M:** Using $y = 3$ at $x = 0$ to find a value for $C$ (dependent on an integration attempt, however poor, on the RHS) | |
**Alternative:**
**1st M:** Multiply through the given equation by $\cos x$ | |
**1st A:** Achieving $\cos x\frac{dy}{dx} - y\sin x = 2\sec^2 x$ (Allowing the possibility of integrating by inspection) | |
**Total: [7]**
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6.
$$\frac { \mathrm { d } y } { \mathrm {~d} x } - y \tan x = 2 \sec ^ { 3 } x$$
Given that $y = 3$ at $x = 0$, find $y$ in terms of $x$\\
(Total 7 marks)\\
\hfill \mbox{\textit{Edexcel FP2 2007 Q6 [7]}}