---
title: "Solve x²y''−2xy'+2y=1/x"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-cauchy-x2y2-minus-2xy1-plus-2y-1-over-x"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-cauchy-x2y2-minus-2xy1-plus-2y-1-over-x
site: BCT Engineering
---

# Solve x²y''−2xy'+2y=1/x

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

- Asked in: 2073 Chaitra
- Video: [27. Non-Homogeneous Linear Equations | Problem#1 | Complete Concept | Most Important Problem | PDE](https://www.youtube.com/watch?v=8OhkbYdxsls) — MKS TUTORIALS by Manoj Sir
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/de-cauchy-x2y2-minus-2xy1-plus-2y-1-over-x
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve: x² d²y/dx² − 2x dy/dx + 2y = 1/x.

## Short answer

Cauchy–Euler equation, transform with z = log x. General solution: y = c₁x + c₂x² + 1/(6x).

## Answer

x²y'' − 2xy' + 2y = 1/x is a Cauchy–Euler equation.

#### Transformation
Put z = log x and D = d/dz, so x²y'' = D(D − 1)y and xy' = Dy:

[D(D − 1) − 2D + 2]y = e^(−z)
(D² − 3D + 2)y = e^(−z), i.e. (D − 1)(D − 2)y = e^(−z)

#### Complementary function
m = 1, 2: CF = c₁eᶻ + c₂e^(2z) = c₁x + c₂x²

#### Particular integral
Put D = −1: PI = e^(−z)/[(−1)² − 3(−1) + 2] = e^(−z)/6 = 1/(6x)

#### General solution
**y = c₁x + c₂x² + 1/(6x)**

## Check yourself

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

1. With z = log x, RHS 1/x becomes:
   A. e^(-z)
   B. e^z
   C. -z
   D. z
   Answer: A. x = e^z so 1/x = e^(-z). (Key point: Transformed equation)

2. CF is:
   A. c1 x + c2 x^2
   B. c1 x^2 + c2 x^3
   C. c1/x + c2/x^2
   D. c1 + c2 x
   Answer: A. m = 1, 2. (Key point: CF)

3. PI for e^(-z) is found with D = -1:
   A. -e^(-z)/6
   B. e^(-z)/3
   C. e^(-z)/2
   D. e^(-z)/6
   Answer: D. 1 + 3 + 2 = 6. (Key point: PI)

4. The solution is:
   A. c1 x + c2 x^2 + 1/(6x)
   B. c1 x + c2 x^2 - 1/(6x)
   C. c1 x + c2 x^2 + x/6
   D. c1 x + c2 x^2 + 1/(2x)
   Answer: A. PI = 1/(6x). (Key point: Final solution)

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