---
title: "Solve y''−2y'+y=x²e^(3x)"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-y2-minus-2y1-plus-y-x2e3x"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-y2-minus-2y1-plus-y-x2e3x
site: BCT Engineering
---

# Solve y''−2y'+y=x²e^(3x)

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

- Asked in: 2071 Shrawan, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/de-y2-minus-2y1-plus-y-x2e3x
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve: d²y/dx² − 2 dy/dx + y = x²e^(3x).

## Short answer

The complementary function is (c₁ + c₂x)eˣ. Using the shift rule, the particular integral is (e^(3x)/4)(x² - 2x + 3/2), so y = (c₁ + c₂x)eˣ + (e^(3x)/4)(x² - 2x + 3/2).

## Answer

The equation is (D² − 2D + 1)y = x²e^(3x).

#### Complementary function
m² − 2m + 1 = 0, so m = 1, 1.

CF = (c₁ + c₂x)eˣ

#### Particular integral
Replace D by D + 3 and take e^(3x) outside:

PI = e^(3x) · 1/(D + 3 − 1)² · x² = e^(3x) · 1/(D + 2)² · x²
 = e^(3x) · (1/4)(1 + D/2)⁻² x²
 = (e^(3x)/4)(1 − D + (3/4)D² − …) x²
 = (e^(3x)/4)(x² − 2x + 3/2)

#### General solution
**y = (c₁ + c₂x)eˣ + (e^(3x)/4)(x² − 2x + 3/2)**

## Check yourself

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

1. m^2 - 2m + 1 = 0 gives CF:
   A. (c1 + c2 x)e^(-x)
   B. (c1 + c2 x)e^x
   C. c1 e^x + c2 e^(-x)
   D. c1 e^x
   Answer: B. Repeated root m = 1. (Key point: Complementary function)

2. For x^2 e^(3x), the operator becomes:
   A. (D - 2)^2
   B. (D + 3)^2
   C. (D + 2)^2
   D. (D - 1)^2
   Answer: C. (D + 3 - 1)^2. (Key point: Shifted operator)

3. (1/4)(1 + D/2)^-2 expands to:
   A. (1/4)(1 + D + (3/4)D^2)
   B. (1/4)(1 - D + (3/4)D^2 - ...)
   C. (1/4)(1 - D + D^2)
   D. (1/4)(1 - D/2 + D^2/4)
   Answer: B. (1 + u)^-2 = 1 - 2u + 3u^2 with u = D/2. (Key point: Expansion)

4. The PI is:
   A. (e^(3x)/4)(x^2 - 2x + 3/2)
   B. e^(3x)(x^2 - 2x + 3/2)
   C. (e^(3x)/4)(x^2 + 2x + 3/2)
   D. (e^(3x)/4)(x^2 - 2x + 3)
   Answer: A. Using (1 - D + (3/4)D^2) on x^2 gives x^2 - 2x + 3/2. (Key point: Final PI)

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