It is given that $x \frac{\mathrm{d}y}{\mathrm{d}x} = xy(\ln x + \ln y) - y$. Using the substituti

Question

It is given that $x \frac{\mathrm{d}y}{\mathrm{d}x} = xy(\ln x + \ln y) - y$. Using the substitution $w = xy$, show that the differential equation can be transformed to $\frac{\mathrm{d}w}{\mathrm{d}x} = \mathrm{f}(w)$, where the function $\mathrm{f}(w)$ is to be found.[3] Hence, given that $y = \frac{1}{2}\mathrm{e}^3$ when $x = 2$, solve the differential equation $x \frac{\mathrm{d}y}{\mathrm{d}x} = xy(\ln x + \ln y) - y$, to find $y$ in terms of $x$.[5]

Accepted Answer

(a) $f(w)=w\ln w$ (b) $y=\frac{e^{3e^{x-2}}}{x}$, with $x>0$

2024 YIJC P1 Q4

2024 YIJC P1 Q4

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