The differential equation of all parabolas whose axes are paralle… - Vidyadip

MCQ

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

✓
$\frac{{{d^3}y}}{{d{x^3}}} = 0$
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

Correct option: A.

$\frac{{{d^3}y}}{{d{x^3}}} = 0$

a
(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 $(i)$ $ w.r.t$. $x$, we get $\frac{{dy}}{{dx}} = 2Ax + B .....(ii)$

Which on differentiating $w.r.t.$ $x$ gives$\frac{{{d^2}y}}{{d{x^2}}} = 2A .....(iii)$

Differentiating $w.r.t.$ $x$ again, we get $\frac{{{d^3}y}}{{d{x^3}}} = 0$.

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