A) \[\frac{{{d}^{3}}y}{d{{x}^{3}}}=0\]
B) \[\frac{{{d}^{2}}x}{d{{y}^{2}}}=c\]
C) \[\frac{{{d}^{3}}y}{d{{x}^{3}}}+\frac{{{d}^{2}}x}{d{{y}^{2}}}=0\]
D) \[\frac{{{d}^{2}}y}{d{{x}^{2}}}+2\frac{dy}{dx}=c\]
Correct Answer: A
Solution :
The equation of a member of the family of parabolas having axis parallel to y-axis is \[y=A{{x}^{2}}+Bx+C\] .....(i) where A, B, C are arbitrary constants. Differentiating (i) w.r.t. x, we get \[\frac{dy}{dx}=2Ax+B\] .....(ii) Which on differentiating w.r.t. x gives\[\frac{{{d}^{2}}y}{d{{x}^{2}}}=2A\] .....(iii) Differentiating w.r.t. x again, we get \[\frac{{{d}^{3}}y}{d{{x}^{3}}}=0\].
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