---
title: "Solve x²y''−4xy'+6y=x"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-cauchy-x2y2-minus-4xy1-plus-6y"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-cauchy-x2y2-minus-4xy1-plus-6y
site: BCT Engineering
---

# Solve x²y''−4xy'+6y=x

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

- Asked in: 2068 Shrawan
- 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-4xy1-plus-6y
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve: x² d²y/dx² − 4x dy/dx + 6y = x.

## Short answer

Cauchy–Euler equation, transform with z = log x. General solution: y = c₁x² + c₂x³ + x/2.

## Answer

x²y'' − 4xy' + 6y = x is a Cauchy–Euler equation.

#### Transformation
Put z = log x, θ = d/dz: x²y'' = θ(θ − 1)y and xy' = θy. The right side is x = eᶻ.

[θ(θ − 1) − 4θ + 6]y = eᶻ
(θ² − 5θ + 6)y = eᶻ, i.e. (θ − 2)(θ − 3)y = eᶻ

#### Complementary function
m = 2, 3: CF = c₁e^(2z) + c₂e^(3z) = c₁x² + c₂x³

#### Particular integral
Put θ = 1: PI = eᶻ/(1 − 5 + 6) = eᶻ/2 = x/2

#### General solution
**y = c₁x² + c₂x³ + x/2**

## Check yourself

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

1. With theta = d/dz, the equation becomes:
   A. (theta^2 + 5theta + 6)y = e^z
   B. (theta^2 - 5theta + 6)y = e^z
   C. (theta^2 - 3theta + 6)y = e^z
   D. (theta^2 - 5theta + 6)y = z
   Answer: B. theta(theta-1) - 4theta + 6. (Key point: Transformed equation)

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

3. PI for e^z (theta = 1):
   A. x/2
   B. x
   C. x/6
   D. x/3
   Answer: A. 1 - 5 + 6 = 2. (Key point: PI)

## Related in Mathematics I

- [Solve (1+y²)+(x−e^(tan⁻¹y))dy/dx=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-1-plus-y2-x-minus-e-atan-y) — Mathematics I, chapter 4, asked 6×
- [Solve dy/dx = y/x + tan(y/x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-dy-dx-y-over-x-tan) — Mathematics I, chapter 4, asked 6×
- [Solve dy/dx − 2y tan x = y² tan x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-bernoulli-2y-tan-x) — Mathematics I, chapter 4, asked 6×
- [Solve (1+x²)dy/dx+y=e^(tan⁻¹x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-1-plus-x2-dy-dx-etan) — Mathematics I, chapter 4, asked 6×
- [Solve x dy/dx + y log y = xy eˣ](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-x-dy-dx-y-log-y) — Mathematics I, chapter 4, asked 4×
- [Solve x log x dy/dx + y = 2 log x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-x-log-x-dy-dx) — Mathematics I, chapter 4, asked 4×

## Related in other subjects

- [Power series: y''−4xy'+(4x²−2)y=0](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Apower-series-y-4xy-4x2) — Mathematics, chapter 4, asked 12×
- [RK-4: dy/dx = (1+x²)y](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Ark4-y-1-x2) — Numerical Method, chapter 6
- [Polar coordinates: ∫∫(x²+y²) dy dx](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Apolar-coordinates-x2-y2-dy) — Mathematics, chapter 2, asked 5×
- [Evaluate ∫∫ e^(y/x) dy dx](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aevaluate-e-y-x-dy) — Mathematics, chapter 2