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

# Solve x²y''−xy'+y=log x

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

- Asked in: 2071 Shrawan, 5 marks; 2067 Asar (asked 2 times)
- Video: [Higher Order Cauchy Euler Equation | Solve x²y''+ xy'+ y=logx sin(logx) | x=e^z Substitution |PYQ #3](https://www.youtube.com/watch?v=j2igllU1UHU) — EP classes by Kiran Sharma
- 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-xy1-plus-y-log-x
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

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

## Short answer

Cauchy–Euler: let z=log x transforms to (D−1)²y=z. CF: (c₁ + c₂ log x)x. PI = z+2. Solution: y = (c₁ + c₂ log x)x + log x + 2.

## Answer

This is a Cauchy–Euler equation. Put z = log x (x = eᶻ) and D = d/dz; then x dy/dx = Dy and x² d²y/dx² = D(D − 1)y.

#### Transformed equation
[D(D − 1) − D + 1]y = z

(D² − 2D + 1)y = z, i.e. (D − 1)²y = z

#### Complementary function
Auxiliary equation (m − 1)² = 0, so m = 1, 1.

CF = (c₁ + c₂z)eᶻ

#### Particular integral
PI = 1/(D − 1)² · z = (1 − D)⁻² z = (1 + 2D + 3D² + …) z = z + 2

#### General solution
y = (c₁ + c₂z)eᶻ + z + 2

Replacing z = log x:

**y = (c₁ + c₂ log x) x + log x + 2**

(Check: y = log x + 2 gives x²y'' − xy' + y = −1 − 1 + log x + 2 = log x ✓.)

## Check yourself

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

1. For a Cauchy-Euler equation, which substitution is used?
   A. z = log x, D = d/dz
   B. z = 1/x
   C. y = vx
   D. z = x^2
   Answer: A. x = e^z turns variable coefficients into constants. (Key point: Substitution)

2. Under z = log x, x^2 d^2y/dx^2 equals:
   A. x D y
   B. D(D + 1)y
   C. D(D - 1)y
   D. D^2 y
   Answer: C. x^2 y'' = D(D-1)y and x y' = Dy. (Key point: Operator forms)

3. The transformed equation (D-1)^2 y = z has CF:
   A. c1 + c2 z
   B. (c1 + c2 z)e^z
   C. e^z(c1 cos z + c2 sin z)
   D. c1 e^z + c2 e^(-z)
   Answer: B. Repeated root m = 1. (Key point: Complementary function)

4. The PI of (D-1)^-2 z is:
   A. 2z
   B. z + 2
   C. z + 1
   D. z - 2
   Answer: B. (1 - D)^-2 z = (1 + 2D + ...) z = z + 2. (Key point: Particular integral)

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