Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 13: Problem 1151

The equation of a curve passing through \([2,(7 / 2)]\) and having gradient \(1-\left(1 / \mathrm{x}^{2}\right)\) at \((\mathrm{x}, \mathrm{y})\) is: (A) \(x y=x+1\) (B) \(x^{2}+x+1\) (C) \(x y=x^{2}+x+1\) (D) none of these

Short Answer

Expert verified
The correct answer is (D) none of these, as none of the given options match the obtained curve equation \[y(x) = x - \frac{1}{x} + 2\].

Step by step solution

01

Integrate the gradient equation

To obtain the equation of the curve from its gradient, we perform integration. We are given the gradient equation as: \[1- \frac{1}{x^2}\] Integrate this equation with respect to x: \[\int(1 - \frac{1}{x^2}) dx\]
02

Evaluate the integral

We have the integral as: \[\int(1 - \frac{1}{x^2}) dx = \int1 dx - \int\frac{1}{x^2} dx\] Now, integrate each term separately: Integral of first term is: \[\int1 dx = x + C_1\] Integral of second term is: \[\int\frac{1}{x^2} dx = -\frac{1}{x} + C_2\] Combining both integrals, we have: \[y(x) = x + C_1 - \frac{1}{x} + C_2\] Let's say, \(C = C_1 + C_2\) (Since, both are constants) So, the equation becomes: \[y(x) = x - \frac{1}{x} + C\]
03

Use the given point to determine the constant

We are given a point \( (2, \frac{7}{2})\) on the curve. Substitute these values into the equation: \[y(2) = 2 - \frac{1}{2} + C\] \[\frac{7}{2} = \frac{3}{2} + C\] Solve for C: \[C = \frac{7}{2} - \frac{3}{2} = 2\]
04

Write the final equation of the curve and choose the correct answer

With the obtained value of C, the final equation of the curve is: \[y(x) = x - \frac{1}{x} + 2\] Let's verify it with the given options: Option (A) is: \[x y = x + 1\] Multiplying both sides of our obtained equation by x, we get: \[x y = x^2 -1 + 2x\] This does not match option (A). Option (B) is an equation in x only, so it doesn't represent a curve with respect to x and y. Option (C) is: \[x y = x^2 + x + 1\] This does not match the equation we obtained. Thus, none of the given options match our obtained curve equation. Therefore, the correct answer is: (D) none of these

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The solution of \(\mathrm{xdy}-\mathrm{ydx}=0\) represents: (A) parabola having vertex at \((0,0)\) (B) circle having centre at \((0,0)\) (C) a straight line passing through \((0,0)\) (D) a rectangular hyperbola

The differential equation of all circles passing through the origin and having their centers on the y-axis is: OR The differential equation for the family of curves \(\mathrm{x}^{2}+\mathrm{y}^{2}-2 \mathrm{ay}=0\), where a is an arbitrary constant is: (A) \(\left(\mathrm{x}^{2}-\mathrm{y}^{2}\right) \mathrm{y}^{1}=2 \mathrm{xy}\) (B) \(2\left(\mathrm{x}^{2}-\mathrm{y}^{2}\right) \mathrm{y}^{1}=\mathrm{xy}\) (C) \(2\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right) \mathrm{y}^{1}=\mathrm{xy}\) (D) \(\left(x^{2}+y^{2}\right) y^{1}=2 x y\)

The order of the differential equation whose general solution is given by \(\mathrm{y}=\mathrm{C}_{1} \mathrm{e}^{(\mathrm{x}+\mathrm{C}) 2}+\left(\mathrm{C}_{3}+\mathrm{C}_{4}\right) \cdot \sin \left(\mathrm{x}+\mathrm{C}_{5}\right)\), where \(\mathrm{C}_{1}\), \(\mathrm{C}_{2}, \mathrm{C}_{3}, \mathrm{C}_{4}, \mathrm{C}_{5}\) are arbitrary Constant is (A) 5 (B) 4 (C) 3 (D) 2

The degree and order of the differential equation of the family of all parabolas whose axis is \(\mathrm{x}\) -axis, are respectively. (A) 1,2 (B) 3,2 (C) 2,3 (D) 2,1

The differential equation of all parabolas having axis parallel to y-axis: (A) \(\left(\mathrm{d}^{3} \mathrm{x} / \mathrm{dy}^{3}\right)^{2}=0\) (B) \(\left(\mathrm{d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)=0\) (C) \(\left(\mathrm{d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)+\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{d} \mathrm{x}^{2}\right)=0\) (D) \(\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)=0\)
See all solutions

Recommended explanations on English Textbooks

Email

Read Explanation

Prosody

Read Explanation

Research and Composition

Read Explanation

Graphology

Read Explanation

History of English Language

Read Explanation

Language and Social Groups

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free