---
title: "Solve (D²−2D+5)y=e^(2x) sin x"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-d2-2d-5-e2x-sinx"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-d2-2d-5-e2x-sinx
site: BCT Engineering
---

# Solve (D²−2D+5)y=e^(2x) sin x

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

- Asked in: 2070 Asar; 2070 Chaitra (asked 2 times)
- 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-e2x-sinx
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve the differential equation (D² − 2D + 5)y = e^(2x) sin x.

## Short answer

CF: eˣ(c₁ cos 2x + c₂ sin 2x). PI: use shift e^(2x)/(D²+2D+5) sin x, then rationalize to (e^(2x)/10)(2 sin x − cos x). General solution: y = eˣ(c₁ cos 2x + c₂ sin 2x) + (e^(2x)/10)(2 sin x − cos x).

## Answer

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

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

CF = eˣ(c₁ cos 2x + c₂ sin 2x)

#### Particular integral
PI = 1/(D² − 2D + 5) · e^(2x) sin x

Replace D by D + 2:

(D+2)² − 2(D+2) + 5 = D² + 2D + 5

PI = e^(2x) · 1/(D² + 2D + 5) · sin x

Put D² = −1: denominator 2D + 4.

= e^(2x) · 1/(2D + 4) · sin x
= e^(2x) · (4 − 2D)/(16 − 4D²) · sin x

With D² = −1, the denominator is 16 + 4 = 20:

= e^(2x)/20 · (4 sin x − 2 cos x)
= (e^(2x)/10)(2 sin x − cos x)

#### General solution
**y = eˣ(c₁ cos 2x + c₂ sin 2x) + (e^(2x)/10)(2 sin x − cos x)**

## Check yourself

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

1. Roots of m^2 - 2m + 5 = 0 give which CF?
   A. c1 e^x + c2 e^(5x)
   B. e^(-x)(c1 cos 2x + c2 sin 2x)
   C. e^(2x)(c1 cos x + c2 sin x)
   D. e^x(c1 cos 2x + c2 sin 2x)
   Answer: D. m = 1 ± 2i. (Key point: Complementary function)

2. Replacing D by D + 2 in D^2 - 2D + 5 gives:
   A. D^2 + 2D + 1
   B. D^2 - 2D + 5
   C. D^2 + 6D + 5
   D. D^2 + 2D + 5
   Answer: D. (D+2)^2 - 2(D+2) + 5 = D^2 + 2D + 5. (Key point: Shifted operator)

3. After D^2 = -1, the operator acting on sin x is:
   A. 2D + 6
   B. 2D + 4
   C. 4 - 2D
   D. D + 4
   Answer: B. D^2 + 2D + 5 becomes -1 + 2D + 5 = 2D + 4. (Key point: Using D^2 = -1)

4. The particular integral is:
   A. (e^(2x)/10)(2 sin x + cos x)
   B. (e^(2x)/10)(2 sin x - cos x)
   C. (e^(2x)/20)(4 sin x + 2 cos x)
   D. (e^(2x)/5)(sin x - cos x)
   Answer: B. (4 - 2D)/20 sin x = (4 sin x - 2 cos x)/20. (Key point: Final PI)

## Related in Mathematics I

- [Solve (D²−2D+5)y=10 sin x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-d2-2d-5-10-sinx) — Mathematics I, chapter 4
- [Solve y''+3y'+2y=e^(2x) sin x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-d2-3d-2-e2x-sinx) — Mathematics I, chapter 4, asked 2×
- [Solve (D²−D−2)y=eˣ+sin 2x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-d2-d-2-ex-sin2x) — Mathematics I, chapter 4
- [Solve (D²+2D+1)y=eˣ cos x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-d2-2d-1-ex-cosx) — Mathematics I, chapter 4, asked 2×
- [Solve (D²+2D+1)y=eˣ+x²](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-d2-2d-1-ex-x2) — Mathematics I, chapter 4
- [Solve dy/dx = (x+2y−3)/(2x+y−3)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-homogeneous-linear-x-2y-3) — Mathematics I, chapter 4

## Related in other subjects

- [Solve a partial differential equation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-a-partial-differential-equation) — Mathematics III, chapter 4
- [Solve a partial differential equation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-a-partial-differential-equation-two) — Mathematics III, chapter 4
- [Solve ODE by Laplace transform x''+2x'+5x=e^(−t)sint](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-x) — Mathematics III, chapter 3, asked 17×
- [Difference equation x(k+2)+3x(k+1)+2x(k)=0](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Adifference-equation-x0-0-x1-1) — Digital Control System, chapter 2, asked 5×