Question

The differential equation of all parabolas whose axis is parallel to the $y-$axis is

• A. $\frac{d^{3}y}{d{x}^3}=0$
• B. $\frac{d^{2}x}{d{y}^2}=0$
• C. $\frac{d^{3}y}{d{x}^3}+\frac{d^{2}x}{d{y}^2}=0$
• D. $\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}=0$

Answer 1

The correct option is A $\frac{d^{3}y}{d{x}^3}=0$

The equation of a member of the family of parabolas having axis parallel to $y-$axis is
$y=A{x}^2+Bx+C\cdots \left(i\right)$
where $A,B$ and $C$ are arbitrary constant
Differentiating equation $\left(i\right)$ w.r.t. $x,$ we get
$\frac{dy}{dx}=2Ax+B\cdots \left(ii\right)$
Which on again differentiating w.r.t. $x$ gives
$\frac{d^{2}y}{d{x}^2}=2A\cdots \left(iii\right)$
Differentiating $\left(iii\right)$ w.r.t. $x,$ we get $\frac{d^{3}y}{d{x}^3}=0$