What is the degree of the differential equation representing the family of curves \(y^{2}=\sqrt{c}\,x\), where \(c\) is a positive parameter?
1
Writing \(k=\sqrt{c}\), the curve is \(y^{2}=kx\); differentiating gives \(2y y'=k\). Substituting back, \(y^{2}=2xy y'\), i.e. \(y=2x y'\), which is a first-degree equation in \(y'\), so the degree is \(1\).
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