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

Hint:
We must first differentiate the parabola's standard equation before further differentiating the resultant equation in order to tackle this particular problem. In this particular question, we shall differentiate the parabola equation three times to arrive at our final conclusion. A parabola is symmetric with its axis. If the equation has ac\[{{\rm{y}}^2}\] term, then the axis of symmetry is along the \[x\]-axis and if the equation has a \[{x^2}\] term, then the axis of symmetry is along the \[y\]-axis.
Formula use:
\[y = {x^n}\]
Differentiate with respect to x
\[\frac{{dy}}{{dx}} = n{x^{n-1}}\]

Complete step-by-step solution:
The equation of the family of parabolas axis parallel to \[y\]-axis having center\[(\alpha ,\beta )\] is given by
\[{(x - \beta )^2} = 4a(y - \alpha ) \ldots \ldots \ldots . \ldots (1)\]
Here,
\[\alpha \],\[\beta \] are two arbitrary constants.
On differentiating the equation (1) with respect to\[x\] by first order derivative, we get
\[2(x - \beta ) = 4a\frac{{dy}}{{dx}}..............(2)\]
On differentiating the equation (2) with respect to\[x\] on both the sides, we get
\[ \Rightarrow 1 = 2a\frac{{{d^2}y}}{{d{x^2}}}...........(3)\]
On differentiating the equation (3) with respect to \[x\] on both sides, we get
\[ \Rightarrow 0 = 2a\frac{{{d^3}y}}{{d{x^3}}}\]
Let’s rearrange the obtained equation by explicitly having all the terms on one side:
\[ \Rightarrow \frac{{{d^3}y}}{{d{x^3}}} = 0\]
Therefore, the differential equation of all parabolas whose axes are parallel to \[y\]-axis is \[\frac{{{d^3}y}}{{d{x^3}}} = 0\]
Hence, the option A is correct.

Note:
Since integration and differentiation are the opposite actions of one another, students can double-check their answers in this question by integrating the final differential equation and determining whether or not it produces the starting equation. Students should also be aware that, in general, we must remove constant terms from an equation in order to determine its differential equation. Student should also remember, for axis of symmetry along the y-axis: It opens upwards if the coefficient of \[y\] is positive. It opens downwards if the coefficient of \[{\rm{y}}\] is negative.