---
title: "Solve y''+2y'=eˣ+e⁻ˣ"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-y2-2y1-e-x-plus-e-minus-x"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-y2-2y1-e-x-plus-e-minus-x
site: BCT Engineering
---

# Solve y''+2y'=eˣ+e⁻ˣ

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

- Asked in: 2069 Chaitra
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/de-y2-2y1-e-x-plus-e-minus-x
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve the differential equation d²y/dx² + 2 dy/dx = eˣ + e⁻ˣ.

## Short answer

The complementary function is c₁ + c₂e^(-2x). The particular integral is eˣ/3 - e^(-x), giving y = c₁ + c₂e^(-2x) + eˣ/3 - e^(-x).

## Answer

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

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

CF = c₁ + c₂e^(−2x)

#### Particular integral
**For eˣ:** put D = 1: eˣ/(1 + 2) = eˣ/3.

**For e^(−x):** put D = −1: e^(−x)/(1 − 2) = −e^(−x).

#### General solution
**y = c₁ + c₂e^(−2x) + eˣ/3 − e^(−x)**

## Check yourself

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

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

2. PI for e^x is:
   A. e^x/2
   B. e^x
   C. -e^x/3
   D. e^x/3
   Answer: D. D = 1 gives 1 + 2 = 3. (Key point: PI for e^x)

3. PI for e^(-x) is:
   A. e^(-x)/3
   B. -e^(-x)
   C. e^(-x)
   D. -e^(-x)/3
   Answer: B. D = -1 gives 1 - 2 = -1. (Key point: PI for e^(-x))

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