Question

The differential equation of all parabolas, whose axes are parallel to $y$ -axis is

• 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. $\frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}=C$

Answer 1

Answer: (A)

The equation of a member of the family of parabolas having axis parallel to $y$ -axis is
$y=A x^{2}+B x+C \,\,\,\,\,\, (i)$
where $A, B, C$ are arbitrary constants. Differentiating (i) with respect to $x,$ we get
$\frac{d y}{d x}=2 A x+B \,\,\,\,\, (ii)$
which on differentiating with respect to $x$ gives
$\frac{d^{2} y}{d x^{2}}=2 A \,\,\,\,\,\, (iii)$
Differentiating (iii) with respect to $x,$ we get $\frac{d^{3} y}{d x^{3}}=0$