---
title: "Solve x y''+2y'=6x"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-xy2-plus-2y1-6x"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-xy2-plus-2y1-6x
site: BCT Engineering
---

# Solve x y''+2y'=6x

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

- Asked in: 2071 Chaitra
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/de-xy2-plus-2y1-6x
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve the differential equation x d²y/dx² + 2 dy/dx = 6x.

## Short answer

Multiply by x to get exact derivative d/dx(x²y') = 6x²; integrate to obtain y' = 2x + c₁/x². General solution: y = x² − c₁/x + c₂.

## Answer

x y'' + 2y' = 6x

#### Method: exact derivative
Multiply both sides by x:

x²y'' + 2xy' = 6x²

The left side is d/dx (x²y'). So

d/dx (x² y') = 6x²

x²y' = 2x³ + c₁

y' = 2x + c₁/x²

#### Integrate again
**y = x² − c₁/x + c₂**

or, writing the constants as C₁ = −c₁ and C₂ = c₂: y = x² + C₁/x + C₂.

(This agrees with the Cauchy–Euler method: with z = log x, (θ² + θ)y = 6e^(2z), CF = c₁ + c₂e^(−z), PI = e^(2z) = x².)

## Check yourself

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

1. Multiplying x y'' + 2y' = 6x by x gives left side equal to:
   A. d/dx(x y')
   B. d/dx(x^2 y')
   C. d^2/dx^2(x y)
   D. d/dx(x^2 y)
   Answer: B. x^2 y'' + 2x y' = (x^2 y')'. (Key point: Trick)

2. After first integration:
   A. y' = 2x^3 + c1
   B. x^2 y' = 6x^3 + c1
   C. x^2 y' = 2x^3 + c1
   D. x y' = 3x^2 + c1
   Answer: C. Integral of 6x^2 is 2x^3. (Key point: First integration)

3. The solution is:
   A. y = x^2 + C1 x + C2
   B. y = x^2 + C1 ln x + C2
   C. y = x^3 + C1/x + C2
   D. y = x^2 + C1/x + C2
   Answer: D. y' = 2x + c1/x^2 integrates to x^2 - c1/x + c2. (Key point: Final solution)

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