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

# Solve x²y''+4xy'+2y=eˣ

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

- Asked in: 2069 Asar; 2068 Chaitra (asked 2 times)
- Video: [x^2d^2y/dx^2-3xdy/dx+5y=x^2Sin(logx) #CauchyEulerLinearEquation L912](https://www.youtube.com/watch?v=H5B0u1hGW54) — Maths Pulse - Chinnaiah Kalpana
- 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-4xy1-plus-2y
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve the differential equation x² d²y/dx² + 4x dy/dx + 2y = eˣ.

## Short answer

Recognise left side as (x²y)''. Integrate twice: (x²y)' = eˣ + c₁, then x²y = eˣ + c₁x + c₂. Hence general solution: y = eˣ/x² + c₁/x + c₂/x².

## Answer

x² y'' + 4x y' + 2y = eˣ

#### Quick method: recognise the left side
Since (x²y)'' = x²y'' + 4xy' + 2y, the equation is

d²(x²y)/dx² = eˣ

Integrate twice:

d(x²y)/dx = eˣ + c₁
x²y = eˣ + c₁x + c₂

**y = eˣ/x² + c₁/x + c₂/x²**

#### Check by the standard (Cauchy–Euler) method
Put z = log x, D = d/dz: (D(D − 1) + 4D + 2)y = e^(eᶻ), i.e. (D + 1)(D + 2)y = e^(eᶻ).

CF = c₁e^(−z) + c₂e^(−2z) = c₁/x + c₂/x², which matches the terms above. The particular integral gives eˣ/x², so the general solution is the same.

## Check yourself

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

1. Which identity lets the equation be written as d^2(x^2 y)/dx^2 = e^x?
   A. (y/x^2)'' = ...
   B. (x^2 y)'' = x^2 y'' + 4x y' + 2y
   C. (x y)'' = x y'' + 2y'
   D. (x^2 y)' = x^2 y' + y
   Answer: B. Expanding (x^2 y)'' by the product rule gives exactly the left side. (Key point: Recognition)

2. After integrating twice, x^2 y equals:
   A. e^x + c1 + c2
   B. e^x/x + c
   C. x e^x + c
   D. e^x + c1 x + c2
   Answer: D. Integrating e^x twice gives e^x + c1 x + c2. (Key point: Integration)

3. The general solution is:
   A. y = e^x + c1/x + c2
   B. y = e^x/x^2 + c1/x + c2/x^2
   C. y = x^2 e^x + c1 x + c2
   D. y = e^x/x + c1 + c2/x
   Answer: B. Divide x^2 y = e^x + c1 x + c2 by x^2. (Key point: Final solution)

4. In the standard method, the CF terms are:
   A. c1/x + c2/x^2
   B. c1 + c2 x
   C. c1/x^2 + c2 x
   D. c1 x + c2 x^2
   Answer: A. (D+1)(D+2) gives e^(-z), e^(-2z) = 1/x, 1/x^2. (Key point: Cross-check)

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