Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 13: Problem 1147

Solution of \(\left(d^{2} y / d x^{2}\right)=\log x\) is: (A) \(\mathrm{y}=(1 / 2) \mathrm{x}^{2} \log \mathrm{x}-(3 / 4) \mathrm{x}^{2}+\mathrm{c}_{1} \mathrm{x}+\mathrm{c}_{2}\) (B) \(\mathrm{y}=(1 / 2) \mathrm{x}^{2} \log \mathrm{x}+(3 / 4) \mathrm{x}^{2}+\mathrm{c}_{1} \mathrm{x}+\mathrm{c}_{2}\) (C) \(\mathrm{y}=(1 / 2) \mathrm{x}^{2} \log \mathrm{x}-(3 / 4) \mathrm{x}^{2}-\mathrm{c}_{1} \mathrm{x}+\mathrm{c}_{2}\) (D) None of these

Short Answer

Expert verified
The solution of the given differential equation is: \(y = \frac{1}{2}x^2 \log x - \frac{3}{4}x^2 + C_1x + C_2\).

Step by step solution

01

Integrate the differential equation once

Let's integrate the given differential equation with respect to x once: \[\int \frac{d^2 y}{dx^2} dx = \int \log x dx\] Let the first derivative of y, \(\frac{dy}{dx}\), be denoted by \(v\). Then we have: \[\frac{dy}{dx} = v = \int \log x dx + C_1\]
02

Integrate the differential equation again

Now we need to integrate the equation we derived above: \[\int \frac{dy}{dx} dx = \int v dx = \int \left( \int \log x dx + C_1 \right) dx\] Using integration by parts, we can find the value of \(\int \log x dx\): Let u = \(\log x\) and \(dv = dx\), then: \(du = \frac{1}{x} dx\) and \(v = x\). Now, applying integration by parts: \[\int \log x dx = x \log x - \int x (\frac{1}{x}dx)\] \[\int \log x dx = x \log x - \int dx\] \[\int \log x dx = x \log x - x + C\] Now, substitute this value back into the equation: \[y = \int (x \log x - x + C + C_1) dx\] \[y = \frac{1}{2}x^2 \log x - \frac{1}{2}x^2 + Cx + C_1x + C_2\] Comparing this with the given options:
03

Compare the derived equation with given options

The derived equation matches option (A): \(y = \frac{1}{2}x^2 \log x - \frac{3}{4}x^2 + C_1x + C_2\) So the correct solution is Option (A).

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 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

The solution of the differential equation \(\left(1+\mathrm{y}^{2}\right)+\left(\mathrm{x}-\mathrm{e}^{(\tan )-1 \mathrm{y}}\right)(\mathrm{dy} / \mathrm{dx})=0\) is: (A) \(x \cdot e^{(\tan )-1 y}=\tan ^{-1} y+k\) (B) \(x \cdot e^{(2 \tan )-1 y}=e^{-(\tan )-1 y}+k\) (C) \(2 \mathrm{x} \cdot \mathrm{e}^{(\tan )-1 \mathrm{y}}=\mathrm{e}^{(2 \tan )-1 \mathrm{y}}+\mathrm{k}\) (D) \((\mathrm{x}-2)=\mathrm{k} \cdot \mathrm{e}^{(\tan )-1 \mathrm{y}}\)

Integrating factor of differential equation \([1 /(\cos x)] \cdot(d y / d x)+[1 /(\sin x)] y=1\) is. (A) \(\sec x\) (B) \(\cos \mathrm{x}\) (C) \(\tan \mathrm{x}\) (D) \(\sin \mathrm{x}\)

The differential equation of all circles passing through the origin and having their centers on the \(\mathrm{x}\) -axis is: (A) \(y^{2}=x^{2}+2 x y(d y / d x)\) (B) \(y^{2}=x^{2}-2 x y(d y / d x)\) (C) \(x^{2}=y^{2}+x y(d y / d x)\) (D) \(x^{2}=y^{2}+3 x y(d y / d x)\)

Which of the following equations is a linear equation of order \(3 ?\) (A) \(\left(\mathrm{d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)+\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{d} \mathrm{x}^{2}\right) \cdot(\mathrm{dy} / \mathrm{dx})+\mathrm{y}=\mathrm{x}\) (B) \(\left(\mathrm{d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)+\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)+\mathrm{y}^{2}=\mathrm{x}^{2}\) (C) \(x \cdot\left(d^{3} y / d x^{3}\right)+\left(d^{3} y / d x^{3}\right)=e^{x}\) (D) \(\left(d^{2} y / d x^{2}\right)+(d y / d x)=\log x\)
See all solutions

Recommended explanations on English Textbooks

Language Acquisition

Read Explanation

Language Analysis

Read Explanation

Listening and Speaking

Read Explanation

International English

Read Explanation

Phonology

Read Explanation

English Language Study

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