---
title: "Solve (D²+2D+1)y=eˣ+x²"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-d2-2d-1-ex-x2"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-d2-2d-1-ex-x2
site: BCT Engineering
---

# Solve (D²+2D+1)y=eˣ+x²

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

- Asked in: 2074 Asoj
- Video: [Solve: (D^2+2D+1)y=2x+x^2 differential equation](https://www.youtube.com/watch?v=NuoxrUPiI4E) — Doctor of Mathematics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/de-d2-2d-1-ex-x2
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve: (D² + 2D + 1)y = eˣ + x².

## Short answer

The complementary function is (c₁ + c₂x)e^(-x). The particular integral is eˣ/4 + x² − 4x + 6, giving general solution y = (c₁ + c₂x)e^(-x) + eˣ/4 + x² − 4x + 6.

## Answer

The equation is (D² + 2D + 1)y = eˣ + x².

#### Complementary function
Auxiliary equation m² + 2m + 1 = 0, so (m + 1)² = 0, m = −1, −1.

CF = (c₁ + c₂x)e^(−x)

#### Particular integral
**For eˣ:** put D = 1:

PI₁ = eˣ/(1 + 2 + 1) = eˣ/4

**For x²:**

PI₂ = 1/(1 + D)² · x² = (1 + D)⁻² x² = (1 − 2D + 3D² − …) x²
 = x² − 2(2x) + 3(2) = x² − 4x + 6

#### General solution
**y = (c₁ + c₂x)e^(−x) + eˣ/4 + x² − 4x + 6**

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

2. PI for e^x is found by D = 1:
   A. e^x/2
   B. e^x/3
   C. e^x/4
   D. x e^x/4
   Answer: C. 1 + 2 + 1 = 4. (Key point: PI for e^x)

3. PI for x^2 uses (1 + D)^-2 expanded as:
   A. 1 - 2D + D^2
   B. 1 - 2D + 3D^2 - ...
   C. 1 - D + D^2
   D. 1 + 2D + 3D^2 + ...
   Answer: B. (1 + D)^-2 = 1 - 2D + 3D^2 - ... (Key point: PI for x^2)

4. Applying it to x^2 gives:
   A. x^2 - 4x + 2
   B. x^2 - 2x + 3
   C. x^2 + 4x + 6
   D. x^2 - 4x + 6
   Answer: D. x^2 - 2(2x) + 3(2). (Key point: Value of PI for x^2)

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