---
title: "Solve (D²−6D+9)y=x²e^(2x)"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-d2-6d-9-x2-e2x"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-d2-6d-9-x2-e2x
site: BCT Engineering
---

# Solve (D²−6D+9)y=x²e^(2x)

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

- Asked in: 2074 Chaitra, 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-d2-6d-9-x2-e2x
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve (D² − 6D + 9)y = x²e^(2x).

## Short answer

The complementary function is (c₁ + c₂x)e^(3x). Using the shift rule, the particular integral is e^(2x)(x² + 4x + 6), so the general solution is y = (c₁ + c₂x)e^(3x) + e^(2x)(x² + 4x + 6).

## Answer

The equation is (D² − 6D + 9)y = x²e^(2x).

#### Complementary function
m² − 6m + 9 = 0, so (m − 3)² = 0, m = 3, 3.

CF = (c₁ + c₂x)e^(3x)

#### Particular integral
Use the shift rule, replacing D by D + 2:

PI = e^(2x) · 1/[(D + 2 − 3)²] · x² = e^(2x) · 1/(D − 1)² · x²
 = e^(2x)(1 − D)⁻² x²
 = e^(2x)(1 + 2D + 3D² + …) x²
 = e^(2x)(x² + 4x + 6)

#### General solution
**y = (c₁ + c₂x)e^(3x) + e^(2x)(x² + 4x + 6)**

## Check yourself

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

1. m^2 - 6m + 9 = 0 gives CF:
   A. c1 e^(3x) + c2 e^(-3x)
   B. (c1 + c2 x)e^(3x)
   C. e^(3x)(c1 cos x + c2 sin x)
   D. (c1 + c2 x)e^(-3x)
   Answer: B. Repeated root m = 3. (Key point: Complementary function)

2. For x^2 e^(2x), D is replaced by D + 2, giving operator:
   A. (D - 1)^2
   B. (D + 1)^2
   C. (D + 2)^2
   D. (D - 3)^2
   Answer: A. (D + 2 - 3)^2 = (D - 1)^2. (Key point: Shift rule)

3. (1 - D)^-2 expands as:
   A. 1 + 2D + 3D^2 + ...
   B. 1 + 2D + D^2
   C. 1 - 2D + 3D^2
   D. 1 + D + D^2
   Answer: A. Binomial series for (1 - D)^-2. (Key point: Expansion)

4. The PI is:
   A. e^(2x)(x^2 + 4x + 6)
   B. e^(2x)(x^2 - 4x + 6)
   C. e^(3x)(x^2 + 4x + 6)
   D. e^(2x)(x^2 + 2x + 2)
   Answer: A. x^2 + 2(2x) + 3(2). (Key point: Final PI)

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