---
title: "Solve (D²−3D+2)y=x²+x"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-d2-3d-2-x2-plus-x"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-d2-3d-2-x2-plus-x
site: BCT Engineering
---

# Solve (D²−3D+2)y=x²+x

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

- Asked in: 2067 Asar
- Video: [Finding the complete solution of (D^2-3D+2)y=x | Second Order Differential Equations | Important ...](https://www.youtube.com/watch?v=LWBQwp5af9U) — SARAVANAKUMAR MATHS
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/de-d2-3d-2-x2-plus-x
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve (D² − 3D + 2)y = x² + x.

## Short answer

The complementary function is c₁eˣ + c₂e^(2x). The particular integral is ½(x² + 4x + 5), so y = c₁eˣ + c₂e^(2x) + ½(x² + 4x + 5).

## Answer

The equation is (D² − 3D + 2)y = x² + x.

#### Complementary function
m² − 3m + 2 = 0, so (m − 1)(m − 2) = 0, m = 1, 2.

CF = c₁eˣ + c₂e^(2x)

#### Particular integral
PI = 1/(2 − 3D + D²) · (x² + x)
 = ½ [1 − (3D/2 − D²/2)]⁻¹ (x² + x)
 = ½ [1 + (3D/2 − D²/2) + (3D/2)² + …] (x² + x)
 = ½ [1 + 3D/2 + (9/4 − 1/2)D² + …] (x² + x)
 = ½ [1 + 3D/2 + (7/4)D²] (x² + x)

Now D(x² + x) = 2x + 1 and D²(x² + x) = 2:

 = ½ [x² + x + (3/2)(2x + 1) + (7/4)(2)]
 = ½ [x² + x + 3x + 3/2 + 7/2]
 = ½ (x² + 4x + 5)

#### General solution
**y = c₁eˣ + c₂e^(2x) + ½(x² + 4x + 5)**

(Check: y_p = ½(x² + 4x + 5); y_p' = x + 2, y_p'' = 1; 1 − 3(x + 2) + (x² + 4x + 5) = x² + x ✓.)

## Check yourself

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

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

2. For PI of a polynomial, the operator is:
   A. Expanded as a power series in D
   B. Set D^2 = -a^2
   C. Replaced by constants
   D. Shifted by D + a
   Answer: A. 1/(2 - 3D + D^2) is expanded in ascending powers of D. (Key point: PI strategy)

3. The coefficient of D^2 inside the bracket is:
   A. 9/4
   B. 1/2
   C. 5/4
   D. 7/4
   Answer: D. (3/2)^2 - 1/2 = 7/4. (Key point: Series coefficient)

4. The PI is:
   A. (1/2)(x^2 - 4x + 5)
   B. (1/2)(x^2 + 2x + 5)
   C. (1/2)(x^2 + 4x + 5)
   D. x^2 + 4x + 5
   Answer: C. Check: y_p'' - 3y_p' + 2y_p = x^2 + x. (Key point: Final PI)

## Related in other subjects

- [Solve ODE by Laplace transform (D²−3D+2)y=e^(3t)underth…](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-d2) — Mathematics III, chapter 3