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

Answer 1

Answer: (A)

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