The variables $$$x$$$ and $$$t$$$ satisfy the differential equation $$$ x \ln x+t \frac{\mathrm{d} x}{\mathrm{~d} t}=0 . $$$ It is given that $$$x=\mathrm{e}$$$ when $$$t=2$$$. Solve the differential equation obtaining an expression for $$$x$$$ in terms of $$$t$$$, simplifying your answer. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ 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Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_w21_qp_31 Year：2021 Question No：7(b)

Answer:

Correct separation of variables
Obtain term $\ln (\ln x)$
Obtain term $-\ln t$
Evaluate a constant or use $x=\mathrm{e}$ and $t=2$ as limits in an expression involving $\ln (\ln x)$
Obtain correct solution in any form, e.g. $\ln (\ln x)=-\ln t+\ln 2$
Use log laws to enable removal of logarithms
Obtain answer $x=\mathrm{e}^{\frac{2}{t}}$, or simplified equivalent

Knowledge points:

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.)

Answer:

Knowledge points:

Solution:

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