---
title: "Solve x²y''+xy'−4y=x²"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-cauchy-x2y2-plus-xy1-minus-4y"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-cauchy-x2y2-plus-xy1-minus-4y
site: BCT Engineering
---

# Solve x²y''+xy'−4y=x²

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

- Asked in: 2072 Kartik
- Video: [Euler Cauchy Equations | Numericals | Ordinary Differential Equations of Higher Order | Maths](https://www.youtube.com/watch?v=IZwKDQfwQsA) — Gautam Varde
- 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-plus-xy1-minus-4y
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve the differential equation x² d²y/dx² + x dy/dx − 4y = x².

## Short answer

Cauchy–Euler equation, transform with z = log x. General solution: y = c₁x² + c₂/x² + (x² log x)/4.

## Answer

x²y'' + xy' − 4y = x² is a Cauchy–Euler equation.

#### Transformation
Put z = log x, D = d/dz:

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

#### Complementary function
m = ±2: CF = c₁e^(2z) + c₂e^(−2z) = c₁x² + c₂x⁻²

#### Particular integral
Since 2 is a root of D² − 4 = 0, use the resonance rule (derivative of the denominator is 2D = 4):

PI = z e^(2z)/(2·2) = (x² log x)/4

#### General solution
**y = c₁x² + c₂/x² + (x² log x)/4**

## 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 - 2D - 4)y = e^(2z)
   B. (D^2 - 4)y = z^2
   C. (D^2 + 4)y = e^(2z)
   D. (D^2 - 4)y = e^(2z)
   Answer: D. D(D-1) + D - 4 = D^2 - 4. (Key point: Transformed equation)

2. CF is:
   A. c1 cos(2 log x) + c2 sin(2 log x)
   B. c1 x^2 + c2 x^-2
   C. c1 x^4 + c2
   D. c1 x + c2 x^2
   Answer: B. m = ±2. (Key point: CF)

3. The resonance rule is used because:
   A. RHS is zero
   B. RHS is polynomial
   C. 2 is a root of D^2 - 4 = 0
   D. Roots are complex
   Answer: C. e^(2z) duplicates a CF term. (Key point: Resonance)

4. PI is:
   A. x^2/4
   B. (x^2 log x)/2
   C. (x^2 log x)/8
   D. (x^2 log x)/4
   Answer: D. z e^(2z)/(2·2). (Key point: PI)

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