---
title: "Solve (D²−D−2)y=eˣ+sin 2x"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-d2-d-2-ex-sin2x"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-d2-d-2-ex-sin2x
site: BCT Engineering
---

# Solve (D²−D−2)y=eˣ+sin 2x

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

- Asked in: 2072 Chaitra
- Video: [Solve: (D²-D-2)y= sin2x | Linear Differential Equation with Constant Coefficient| CF and PI](https://www.youtube.com/watch?v=j1tnAWP4_Nw) — RK Eduworld (Classic)
- 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-d-2-ex-sin2x
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve: (D² − D − 2)y = eˣ + sin 2x.

## Short answer

The complementary function is c₁e^(2x) + c₂e^(-x). The particular integral is -eˣ/2 - (3 sin 2x - cos 2x)/20, giving y = c₁e^(2x) + c₂e^(-x) - eˣ/2 - (3 sin 2x - cos 2x)/20.

## Answer

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

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

CF = c₁e^(2x) + c₂e^(−x)

#### Particular integral
**For eˣ:** put D = 1: 1/(1 − 1 − 2) eˣ = −eˣ/2.

**For sin 2x:** put D² = −4:

1/(D² − D − 2) sin 2x = 1/(−6 − D) sin 2x = −1/(D + 6) sin 2x
 = −(D − 6)/(D² − 36) sin 2x

With D² = −4 the denominator is −40:

 = −(D − 6)/(−40) sin 2x = (D − 6) sin 2x/40 = (2 cos 2x − 6 sin 2x)/40
 = −(3 sin 2x − cos 2x)/20

#### General solution
**y = c₁e^(2x) + c₂e^(−x) − eˣ/2 − (3 sin 2x − cos 2x)/20**

## Check yourself

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

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

2. PI for e^x is:
   A. e^x/4
   B. e^x/2
   C. -e^x/2
   D. -e^x/4
   Answer: C. D = 1 gives 1 - 1 - 2 = -2. (Key point: PI for e^x)

3. For sin 2x we put:
   A. D = -2
   B. D^2 = -4
   C. D^2 = 4
   D. D = 2
   Answer: B. For sin ax, D^2 = -a^2. (Key point: PI for sin 2x)

4. The PI for sin 2x is:
   A. -(3 sin 2x - cos 2x)/20
   B. -(3 sin 2x + cos 2x)/20
   C. (3 sin 2x - cos 2x)/20
   D. (sin 2x)/6
   Answer: A. Denominator -6 - D leads to (D - 6) sin 2x/40. (Key point: Final PI of sin 2x)

## Related in Mathematics I

- [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 (D²−2D+5)y=e^(2x) sin x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-d2-2d-5-e2x-sinx) — Mathematics I, chapter 4, asked 2×
- [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
- [Solve (D²−6D+9)y=x²e^(2x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-d2-6d-9-x2-e2x) — Mathematics I, chapter 4
- [Integral of (sin x+cos x)/(9+16 sin 2x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aintegral-sinx-cosx-over-9-16-sin2x) — Mathematics I, chapter 2
- [Solve sin x dy/dx + y cos x = x sin x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-sinx-dy-dx-y-cosx) — Mathematics I, chapter 4

## Related in other subjects

- [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×
- [Composite Simpson 3/8: e^x + sin 2x](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Asimpson-38-composite) — Numerical Method, chapter 5
- [Gauss 2-point and 3-point: log(x+1)+sin 2x](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Agauss-2pt-3pt-log-sin) — Numerical Method, chapter 5
- [Simplex method max:Z=10x₁+x₂+2x₃](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asimplex-method-max-z-10x-x) — Mathematics III, chapter 5, asked 18×