Consider the differential equation \(\frac{\mathrm{d}y}{\mathrm{d}x} = \mathrm{f}\left( \frac{y}{x}

Question

Consider the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x} = \mathrm{f}\left( \frac{y}{x} \right), \quad x > 0.$ Use the substitution $y = vx$ to show that the general solution of this differential equation is $\displaystyle\int \frac{\mathrm{d}v}{\mathrm{f}(v) - v} = \ln x + \text{Constant}.$[3] Hence, or otherwise, solve the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x^2 + 3xy + y^2}{x^2}, \quad x > 0,$ given that $y = 1$ when $x = 1$. Give your answer in the form $y = \mathrm{g}(x)$.[10]

Accepted Answer

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

2017 IB HL P3 Q4

2017 IB HL P3 Q4

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