The variables $$\(x\)$$ and $$\(y\)$$ satisfy the differential equation $$\( (1-\cos x) \frac{\mathrm{d} y}{\mathrm{~d} x}=y \sin x . \)$$ It is given that $$\(y=4\)$$ when $$\(x=\pi\)$$. Solve the differential equation, obtaining an expression for $$\(y\)$$ in terms of $$\(x\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ 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Exam No：9709_m21_qp_32 Year：2021 Question No：4(a)
Answer:
Separate variables correctly and attempt integration of at least one side
Obtain term \(\ln y\)
Obtain term of the form \(\pm \ln (1-\cos x)\)
Obtain term \(\ln (1-\cos x)\)
Use \(x=\pi, y=4\) to evaluate a constant, or as limits, in a solution containing terms of the form \(a \ln y\) and \(b \ln (1-\cos x)\)
Obtain final answer \(y=2(1-\cos x)\)
Obtain term \(\ln y\)
Obtain term of the form \(\pm \ln (1-\cos x)\)
Obtain term \(\ln (1-\cos x)\)
Use \(x=\pi, y=4\) to evaluate a constant, or as limits, in a solution containing terms of the form \(a \ln y\) and \(b \ln (1-\cos x)\)
Obtain final answer \(y=2(1-\cos x)\)
Knowledge points:
3.2.1 understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base)
3.5.4 recognise an integrand of the form , and integrate such functions
3.8.2 find by integration a general form of solution for a first order differential equation in which the variables are separable (Including any of the integration techniques from topic 3.5 above.)
3.8.3 use an initial condition to find a particular solution
Solution:
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