---
title: "Solve y''+3y'+2y=e^(2x) sin x"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-d2-3d-2-e2x-sinx"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-d2-3d-2-e2x-sinx
site: BCT Engineering
---

# Solve y''+3y'+2y=e^(2x) sin x

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

- Asked in: 2073 Shrawan; 2071 Chaitra (asked 2 times)
- Video: [11. CF and PI | Complete Concept & Problem#1 |Most Important| Differential Equations of Higher Order](https://www.youtube.com/watch?v=S0ZKVzCVMsQ) — MKS TUTORIALS by Manoj Sir
- 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-e2x-sinx
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve: d²y/dx² + 3 dy/dx + 2y = e^(2x) sin x.

## Short answer

CF: c₁e^(−x)+c₂e^(−2x). PI: shift D→D+2, then rationalize to get (e^(2x)/170)(11 sin x − 7 cos x). General solution: y = c₁e^(−x) + c₂e^(−2x) + (e^(2x)/170)(11 sin x − 7 cos x).

## Answer

The equation is (D² + 3D + 2)y = e^(2x) sin x, with D = d/dx.

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

CF = c₁e^(−x) + c₂e^(−2x)

#### Particular integral (PI)
PI = 1/(D² + 3D + 2) · e^(2x) sin x

Use the shift rule: replace D by D + 2 and take e^(2x) out:

PI = e^(2x) · 1/[(D+2)² + 3(D+2) + 2] · sin x
   = e^(2x) · 1/(D² + 7D + 12) · sin x

Put D² = −1:

= e^(2x) · 1/(7D + 11) · sin x
= e^(2x) · (11 − 7D)/(121 − 49D²) · sin x

With D² = −1 the denominator is 121 + 49 = 170:

= e^(2x)/170 · (11 sin x − 7 cos x)

#### General solution
**y = c₁e^(−x) + c₂e^(−2x) + (e^(2x)/170)(11 sin x − 7 cos x)**

## Check yourself

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

1. The roots of m^2 + 3m + 2 = 0 give which CF?
   A. c1 e^(-x) + c2 e^(-2x)
   B. e^(-x)(c1 cos x + c2 sin x)
   C. (c1 + c2 x)e^(-x)
   D. c1 e^x + c2 e^(2x)
   Answer: A. m = -1, -2 are real and distinct. (Key point: Complementary function)

2. To find PI for e^(2x) sin x, what replacement is made?
   A. D^2 by -4
   B. D by 2
   C. D by D - 2
   D. D by D + 2, e^(2x) taken outside
   Answer: D. For e^(ax)V, replace D by D + a. (Key point: Shift rule)

3. (D+2)^2 + 3(D+2) + 2 simplifies to:
   A. D^2 + 5D + 6
   B. D^2 + 7D + 4
   C. D^2 + 3D + 2
   D. D^2 + 7D + 12
   Answer: D. D^2 + 4D + 4 + 3D + 6 + 2 = D^2 + 7D + 12. (Key point: Shifted operator)

4. The particular integral is:
   A. (e^(2x)/11)(sin x - cos x)
   B. (e^(2x)/12)(sin x)
   C. (e^(2x)/170)(11 sin x + 7 cos x)
   D. (e^(2x)/170)(11 sin x - 7 cos x)
   Answer: D. With D^2 = -1: 1/(7D + 11) = (11 - 7D)/(121 + 49) and D sin x = cos x. (Key point: Final PI)

## Related in Mathematics I

- [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 dy/dx − 2y tan x = y² tan x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-bernoulli-2y-tan-x) — Mathematics I, chapter 4, asked 6×
- [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
- [Solve (1+y²)+(x−e^(tan⁻¹y))dy/dx=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-1-plus-y2-x-minus-e-atan-y) — Mathematics I, chapter 4, asked 6×
- [Solve dy/dx = y/x + tan(y/x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-dy-dx-y-over-x-tan) — Mathematics I, chapter 4, asked 6×
- [Solve (1+x²)dy/dx+y=e^(tan⁻¹x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-1-plus-x2-dy-dx-etan) — Mathematics I, chapter 4, asked 6×

## Related in other subjects

- [Solve ODE by Laplace transform y''+2y'−3y=sint](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y) — Mathematics III, chapter 3, asked 14×
- [Solve ODE by Laplace transform y''+3y'+2y=e^(−t)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-two) — Mathematics III, chapter 3, asked 18×
- [Solve ODE by Laplace transform y''−3y'+2y=4x+e^(3x)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-ten) — Mathematics III, chapter 3, asked 14×
- [Solve ODE by Laplace transform y''+3y'+2y=eᵗ](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-three) — Mathematics III, chapter 3, asked 4×