---
title: "Solve (D²−2D+5)y=10 sin x"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-d2-2d-5-10-sinx"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-d2-2d-5-10-sinx
site: BCT Engineering
---

# Solve (D²−2D+5)y=10 sin x

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

- Asked in: 2068 Chaitra
- Video: [(d^2-2d+5)y=e^2x sin x || Higher order linear differential equation || #differentialequation](https://www.youtube.com/watch?v=AJwyksYY5X0) —  POINT STUDY CENTRE 
- 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-5-10-sinx
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve (D² − 2D + 5)y = 10 sin x.

## Short answer

The complementary function is eˣ(c₁ cos 2x + c₂ sin 2x). The particular integral is 2 sin x + cos x, so the general solution is y = eˣ(c₁ cos 2x + c₂ sin 2x) + 2 sin x + cos x.

## Answer

The equation is (D² − 2D + 5)y = 10 sin x.

#### Complementary function
m² − 2m + 5 = 0, so m = 1 ± 2i.

CF = eˣ(c₁ cos 2x + c₂ sin 2x)

#### Particular integral
Put D² = −1:

PI = 10/(D² − 2D + 5) · sin x = 10/(4 − 2D) · sin x
 = 10(4 + 2D)/(16 − 4D²) · sin x

With D² = −1 the denominator is 20:

 = (4 + 2D) sin x / 2 = (4 sin x + 2 cos x)/2 = 2 sin x + cos x

#### General solution
**y = eˣ(c₁ cos 2x + c₂ sin 2x) + 2 sin x + cos x**

## Check yourself

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

1. m^2 - 2m + 5 = 0 gives m =
   A. 1 ± 4i
   B. 1 ± 2i
   C. -1 ± 2i
   D. 2 ± i
   Answer: B. m = (2 ± sqrt(4 - 20))/2. (Key point: Complementary function)

2. For sin x we put:
   A. D^2 = -1
   B. D^2 = -5
   C. D = -1
   D. D^2 = 1
   Answer: A. For sin ax, D^2 = -a^2. (Key point: Replacement)

3. The denominator 4 - 2D is rationalised by multiplying by:
   A. D + 2
   B. 4 - 2D
   C. 4 + 2D
   D. 2D
   Answer: C. (4 - 2D)(4 + 2D) = 16 - 4D^2 = 20. (Key point: Rationalising)

4. The PI is:
   A. sin x + 2 cos x
   B. 2 sin x - cos x
   C. 2 sin x + cos x
   D. 10 sin x
   Answer: C. (4 sin x + 2 cos x)/2. (Key point: Final PI)

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