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 :
[a] The equation of a member of the family of parabolas having axis parallel to y-axis is \[y=A{{x}^{2}}+Bx+C\] ...(1) Where A, B, and C are arbitrary constants. Differentiating equation (1) w.r.t. x, we get \[\frac{dy}{dx}=2Ax+B\] ...(2) Which on again differentiating w.r.t. x gives \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=2A\] ...(3) Differentiating (3) w.r.t. x, we get \[\frac{{{d}^{3}}y}{d{{x}^{3}}}=0\]
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