---
title: "Solve x²y''−2xy'−4y=x⁴"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-cauchy-x2y2-minus-2xy1-minus-4y"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-cauchy-x2y2-minus-2xy1-minus-4y
site: BCT Engineering
---

# Solve x²y''−2xy'−4y=x⁴

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

- Asked in: 2074 Asoj
- 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-minus-4y
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve: x² d²y/dx² − 2x dy/dx − 4y = x⁴.

## Short answer

Cauchy–Euler equation, transform with z = log x. General solution: y = c₁x⁴ + c₂/x + (x⁴ log x)/5.

## Answer

x²y'' − 2xy' − 4y = x⁴ is a Cauchy–Euler equation.

#### Transformation
Put z = log x, so x = eᶻ, and D = d/dz. Then x y' = Dy and x² y'' = D(D − 1)y:

[D(D − 1) − 2D − 4]y = e^(4z)
(D² − 3D − 4)y = e^(4z), i.e. (D − 4)(D + 1)y = e^(4z)

#### Complementary function
m = 4, −1: CF = c₁e^(4z) + c₂e^(−z) = c₁x⁴ + c₂/x

#### Particular integral
The right side e^(4z) has the same exponent as a root, so use the resonance rule:

PI = z e^(4z) / (2·4 − 3) = z e^(4z)/5 = (x⁴ log x)/5

#### General solution
**y = c₁x⁴ + c₂/x + (x⁴ log x)/5**

## Check yourself

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

1. With z = log x, the equation becomes:
   A. (D^2 - 3D - 4)y = z^4
   B. (D^2 - 3D - 4)y = e^(4z)
   C. (D^2 - D - 4)y = e^(4z)
   D. (D^2 + D - 4)y = e^z
   Answer: B. D(D-1) - 2D - 4. (Key point: Transformed equation)

2. CF is:
   A. c1 x^4 + c2/x
   B. c1 x^4 + c2 x^(-4)
   C. c1 x^4 + c2 x
   D. c1 x^(-4) + c2 x
   Answer: A. m = 4, -1. (Key point: CF)

3. Why does PI take the form z e^(4z)/5?
   A. The roots are complex
   B. 4 is a root of the auxiliary equation
   C. D = 0 is a root
   D. The RHS is polynomial
   Answer: B. e^(4z) matches the CF term, so multiply by z and divide by derivative 2m - 3 = 5. (Key point: Resonance)

4. The general solution is:
   A. c1 x^4 + c2/x + x^4/5
   B. c1 x^4 + c2/x + (x^4 log x)/5
   C. c1 x^4 + c2/x + x^4 log x
   D. c1 x^4 + c2/x + (x^4 log x)/3
   Answer: B. PI = z e^(4z)/5. (Key point: Final solution)

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