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Chapter 13: Problem 1102

The order of differential equation of all parabola with it's axis paralled to \(\mathrm{y}\) -axis and touch \(\mathrm{x}\) -axis is. (A) 2 (B) 3 (C) 1 (D) none of these

Short Answer

Expert verified
The order of the differential equation representing all parabolas with their axis parallel to the y-axis and touching the x-axis is 1.

Step by step solution

01

Find the General Equation of the Parabola

For a parabola with its axis parallel to the y-axis and touching the x-axis, the general equation can be written as: \(x-h = p(y-k)^2\) Here, (h, k) is the vertex of the parabola, and p is the distance between the vertex and the focus (or the directrix). Since the parabola touches the x-axis, its vertex is (h, 0). So, the general equation becomes: \(x-h = p(y-0)^2\) or \(x = h + py^2\)
02

Differentiate the Equation w.r.t y

Now we need to differentiate the equation with respect to y to find the differential equation for the given parabolas: \(\frac{d}{dy}(x) = \frac{d}{dy}(h + py^2)\) As h is constant and independent of y, its differential will be zero: \(\frac{dx}{dy} = 2py\)
03

Eliminate the Parameter p

We have one parameter p in the equation, which we need to eliminate to obtain the differential equation: From the equation of the parabola, we have: \(p = \frac{x - h}{y^2}\) Substitute this expression for p into the equation we obtained after differentiation: \(\frac{dx}{dy} = 2(\frac{x - h}{y^2})y\) Simplify the equation: \(\frac{dx}{dy} = 2\frac{x-h}{y}\)
04

Determine the Order of the Differential Equation

The order of a differential equation is the highest order of derivative present in the equation. In our case, we have only one derivative \(\frac{dx}{dy}\), which is the first derivative. So, the order of the differential equation is 1. Therefore, the correct answer is: (C) 1

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Most popular questions from this chapter

If the general solution of \((\mathrm{dy} / \mathrm{dx})=(\mathrm{y} / \mathrm{x})+\mathrm{f}(\mathrm{x} / \mathrm{y})\) is \(\mathrm{y}\) \(=[\mathrm{x} / \log |\mathrm{cx}|]\), then \(\mathrm{f}(\mathrm{x} / \mathrm{y})\) is given by: (A) \(\left(\mathrm{x}^{2} / \mathrm{y}^{2}\right)\) (B) \(\left(\mathrm{y}^{2} / \mathrm{x}^{2}\right)\) (C) \(\left[\left(-\mathrm{x}^{2}\right) / \mathrm{y}^{2}\right]\) (D) \(\left[\left(-\mathrm{y}^{2}\right) / \mathrm{x}^{2}\right]\)

The differential equation of all conics having centre at the origin is of order. (A) 2 (B) 3 (C) 4 (D) 5

The differential equation whose solution is \(\mathrm{Ax}^{2}+\mathrm{By}^{2}=1\), where \(\mathrm{A}\) and \(\mathrm{B}\) are arbitrary constants is of. (A) second order and second degree (B) first order and first degree (C) first order and second degree (D) second order and first degree

The solution of \(\mathrm{x}^{3}(\mathrm{dy} / \mathrm{dx})+4 \mathrm{x}^{2} \cdot \tan \mathrm{y}=\mathrm{e}^{\mathrm{x}} \cdot \mathrm{sec} \mathrm{y}\) satisfying \(\mathrm{y}(1)=0\) is: (A) \(\sin \mathrm{y}=\mathrm{e}^{\mathrm{x}}(\mathrm{x}-1) \mathrm{x}^{-4}\) (B) \(\tan \mathrm{y}=(\mathrm{x}-1) \mathrm{e}^{\mathrm{x}} \cdot \mathrm{x}^{-3}\) (C) \(\sin \mathrm{y}=\mathrm{e}^{\mathrm{x}}(\mathrm{x}-1) \mathrm{x}^{-3}\) (D) \(\tan \mathrm{y}=(\mathrm{x}-2) \mathrm{e}^{\mathrm{x}} \cdot \log \mathrm{x}\)

If \(y=\left[x+\sqrt{ \left.\left(1+x^{2}\right)\right]^{n}}\right.\), then \(\left(1+x^{2}\right) \cdot\left(d^{2} y / d x^{2}\right)+x \cdot(d y / d x)\) - (A) - (B) \(2 \mathrm{x}^{2} \mathrm{y}\) (C) \(\mathrm{n}^{2} \mathrm{y}\) (D) \(-\mathrm{n}^{2} \mathrm{y}\)
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