Question

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

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

Answer 1

Answer: (A)

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