Homogeneous first-order differential equation solved by substituting y equals vx, with an initial condition

Junior College 2

13 marks

A differential equation is given by
$\frac{\mathrm{d}y}{\mathrm{d}x} = \mathrm{f}\left( \frac{y}{x} \right), \quad x > 0.$
By writing $y = vx$, show that the general solution of this equation can be expressed as
$\displaystyle\int \frac{\mathrm{d}v}{\mathrm{f}(v) - v} = \ln x + \text{Constant}.$[3]
Hence, or by another method, find the solution of the differential equation
$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x^2 + 3xy + y^2}{x^2}, \quad x > 0,$
for which $y = 1$ at $x = 1$, expressing your answer in the form $y = \mathrm{g}(x)$.[10]

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Answer:(a) $\displaystyle\int\dfrac{\mathrm dv}{\mathrm f(v)-v}=\ln x+C$ (b) $y=\dfrac{x(1+2\ln x)}{1-2\ln x}$

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