---
title: "Solve x log x dy/dx + y = 2 log x"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-x-log-x-dy-dx"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-x-log-x-dy-dx
site: BCT Engineering
---

# Solve x log x dy/dx + y = 2 log x

Mathematics I · Chapter 4: Ordinary Differential Equations and their Applications

- Asked in: 2072 Chaitra
- Video: [Solve: x logx dy/dx + y = 2/x logx | NCERT Ex-9.6 Q7](https://www.youtube.com/watch?v=Zi7R_2raJzQ) — RK Eduworld (Classic)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/de-x-log-x-dy-dx
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve the differential equation: x log x dy/dx + y = 2 log x.

## Short answer

Divide by x log x to get linear form with integrating factor log x. Multiplying and integrating (log x)² gives the solution y = log x + c/log x.

## Answer

x log x dy/dx + y = 2 log x

#### Standard linear form
Divide by x log x:

dy/dx + y/(x log x) = 2/x

P = 1/(x log x), Q = 2/x.

#### Integrating factor
∫ dx/(x log x) = log(log x), so IF = log x.

#### Solution
y log x = ∫ (2/x) log x dx = (log x)² + c

(since ∫ (log x)/x dx = (log x)²/2)

**y = log x + c/log x**

(Check: y' = 1/x − c/(x (log x)²); x log x · y' = log x − c/log x; adding y = log x + c/log x gives 2 log x ✓.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Dividing by x log x, P and Q are:
   A. P = x log x, Q = 2 log x
   B. P = 1/(x log x), Q = 2/x
   C. P = 1/x, Q = 2 log x
   D. P = 1/(x log x), Q = 2 log x
   Answer: B. 2 log x/(x log x) = 2/x. (Key point: Standard form)

2. The integrating factor is:
   A. 1/log x
   B. log x
   C. x log x
   D. log(log x)
   Answer: B. Integral of dx/(x log x) = log(log x), so IF = log x. (Key point: Integrating factor)

3. Integral of (2/x) log x dx is:
   A. 2 log x + c
   B. (log x)^2/2 + c
   C. 2 x log x + c
   D. (log x)^2 + c
   Answer: D. Integral of (log x)/x dx = (log x)^2/2, times 2. (Key point: Right-hand integral)

4. The solution is:
   A. y = log x + c log x
   B. y = 2 log x + c
   C. y = (log x)^2 + c
   D. y = log x + c/log x
   Answer: D. Divide y log x = (log x)^2 + c by log x. (Key point: Final solution)

## Close variants set in other exams

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