Discussion
The general solution of the differential equation
\(x \sin(\frac{y}{x}) \frac{dy}{dx} = y \sin(\frac{y}{x}) + \frac{x}{2}\), \(x \gt 0\)
is given by ______.
Answer: D
The general solution is \(\cos(\frac{y}{x}) + \log_{e}(\sqrt{x}) = K\).
The equation is homogeneous, so substitute \(y = vx\), which gives \(\frac{dy}{dx} = v + xv'\):
\[ \begin{aligned} x \sin v\,(v + xv') &= vx \sin v + \frac{x}{2} \\ x^{2} \sin v\,v' &= \frac{x}{2} \\ \sin v\,dv &= \frac{dx}{2x} \end{aligned} \]
Integrating both sides:
\[ \begin{aligned} -\cos v &= \tfrac{1}{2}\log_{e} x + C \\ \cos(\tfrac{y}{x}) + \log_{e}\sqrt{x} &= K \end{aligned} \]
The options with \(\cos(\frac{x}{y})\) invert the substitution variable. The options with \(\log_{e}(x^{2})\) come from writing \(\frac{dx}{2x}\) as \(\frac{2\,dx}{x}\), turning the factor one half into two.
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