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Chapter 13: Problem 1121

The differential equation of family of parabolas with focus at origin and \(\mathrm{x}\) -axis as axis is: (A) \(\mathrm{y}(\mathrm{dy} / \mathrm{dx})^{2}-2 \mathrm{x}(\mathrm{dy} / \mathrm{dx})=\mathrm{y}\) (B) \(\mathrm{y}(\mathrm{dy} / \mathrm{dx})^{2}+2 \mathrm{xy}(\mathrm{dy} / \mathrm{dx})=\mathrm{y}\) (C) \(\mathrm{y}(\mathrm{dy} / \mathrm{dx})^{2}-2 \mathrm{xy}(\mathrm{dy} / \mathrm{dx})=\mathrm{y}\) (D) \(\mathrm{y}(\mathrm{dy} / \mathrm{dx})^{2}+2 \mathrm{x}(\mathrm{dy} / \mathrm{dx})=\mathrm{y}\)

Short Answer

Expert verified
The short answer is: (A) \(y\left(\frac{dy}{dx}\right)^2 -2x\frac{dy}{dx} = y\).

Step by step solution

01

Write a general equation for the parabolas

The general equation of a parabola with its focus at (0, 0), and x-axis as the axis is given by: \[ x^2 = 4ay \]
02

Differentiate the equation with respect to x

Let's differentiate both sides of the equation \( x^2 = 4ay \) with respect to x to get: \[ 2x = 4a \frac{dy}{dx} \]
03

Solve for 'a'

We can solve for 'a' from the above equation, in terms of the derivative \(\frac{dy}{dx}\): \[ a = \frac{x}{2}\frac{dx}{dy} \]
04

Replace 'a' in the original equation

Substitute the value of 'a' into the original equation \( x^2 = 4ay \): \[ x^2 = 4\left(\frac{x}{2}\frac{dx}{dy}\right)y \]
05

Simplify the equation to obtain the differential equation

Simplifying the above equation yields: \[ x^2 = 2xy\frac{dy}{dx} \] Now, we need to rearrange the equation to get it in a form similar to the given options: \[ y\left(\frac{dy}{dx}\right)^2 -2x\frac{dy}{dx} = y \] This differential equation matches option (A), so the correct answer is: (A) \(y\left(\frac{dy}{dx}\right)^2 -2x\frac{dy}{dx} = y\)

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Most popular questions from this chapter

The curve for which the slop of the tangent at any point equals the ratio of the abscissa to the ordinate of the point is: (A) a circle (B) an ellipse (C) a rectangular hyperbola (D) none of these

Order and degree of differential equation of all tangent lines to the parabola \(\mathrm{y}^{2}=4 \mathrm{ax}\) is (A) 2,2 (B) 3,1 (C) 1,2 (D) 4,1

Differential equation of the curves having the subnormal with \((7 / 2)\) units and passes through \((0,0)\) is: (A) \(x^{2}=7 y\) (B) \(\mathrm{y}^{2}=7 \mathrm{x}+\mathrm{c}, \mathrm{c} \neq 0\) (C) \(y^{2}=7 x\) (D) None of these

The solution of the differential equation \((\mathrm{dy} / \mathrm{dx})=(\mathrm{y} / \mathrm{x})+\left[\\{\Phi(\mathrm{y} / \mathrm{x})\\} /\left\\{\Phi^{1}(\mathrm{y} / \mathrm{x})\right\\}\right]\) is: (A) \(\phi(\mathrm{y} / \mathrm{x})=\mathrm{kx}\) (B) \(\Phi(\mathrm{y} / \mathrm{x})=\mathrm{ky}\) (C) \(\mathrm{x} \cdot \Phi(\mathrm{y} / \mathrm{x})=\mathrm{k}\) (D) \(\mathrm{y} \cdot \Phi(\mathrm{y} / \mathrm{x})=\mathrm{k}\)

Solution of \((\mathrm{dy} / \mathrm{dx})=1+\mathrm{x}+\mathrm{y}^{2}+\mathrm{xy}^{2}, \mathrm{y}(0)=0\) is: (A) \(y=\tan \left(c+x+x^{2}\right)\) (B) \(\mathrm{y}=\tan \left[\mathrm{x}+\left(\mathrm{x}^{2} / 2\right)\right]\) (C) \(\mathrm{y}^{2}=\exp \left[\mathrm{x}+\left(\mathrm{x}^{2} / 2\right)\right]\) (D) \(\mathrm{y}^{2}=1+\mathrm{c} \cdot \exp \left[\mathrm{x}+\left(\mathrm{x}^{2} / 2\right)\right]\)
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