---
title: "Solve ODE by Laplace transform y''+3y'+2y=eᵗ"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:solve-ode-by-laplace-transform-y-three"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/solve-ode-by-laplace-transform-y-three
site: BCT Engineering
---

# Solve ODE by Laplace transform y''+3y'+2y=eᵗ

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2082 Shrawan, 5 marks; 2081 Baishakh, 5 marks (asked 2 times)
- Video: [Solving a Second Order IVP By Laplace Transform | y^''-3y^'+2y=e^(-4t),       y(0)=1    y^' (0)=5](https://www.youtube.com/watch?v=y9WPraSR9WI) — math experts
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/solve-ode-by-laplace-transform-y-three
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Solve the differential equation y'' + 3y' + 2y = eᵗ, y(0) = 0 = y'(0) by the Laplace transform method.

## Short answer

Transform gives Y = 1/[(s-1)(s+1)(s+2)]. Partial fractions and inverse transform yield y(t) = e^t/6 - e^(-t)/2 + e^(-2t)/3.

## Answer

Take the Laplace transform of y'' + 3y' + 2y = eᵗ with y(0) = y'(0) = 0, Y = L{y}.

```text
(s² + 3s + 2) Y = 1/(s − 1)
Y = 1/[(s − 1)(s + 1)(s + 2)]
```

Partial fractions: Y = A/(s − 1) + B/(s + 1) + C/(s + 2).

```text
A = 1/[(1 + 1)(1 + 2)] = 1/6
B = 1/[(−1 − 1)(−1 + 2)] = −1/2
C = 1/[(−2 − 1)(−2 + 1)] = 1/3
```

Inverse transform:

```text
y(t) = (1/6) eᵗ − (1/2) e^(−t) + (1/3) e^(−2t)
```

**Answer: y(t) = eᵗ/6 − e^(−t)/2 + e^(−2t)/3.** Check: y(0) = 1/6 − 1/2 + 1/3 = 0 ✓; y'(0) = 1/6 + 1/2 − 2/3 = 0 ✓.

## Check yourself

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

1. Y=
   A. 1/[(s+1)(s+2)]
   B. 1/[(s-1)(s-2)]
   C. 1/[(s-1)(s+1)(s+2)]
   D. 1/[(s+1)^2(s+2)]
   Answer: C. RHS transform 1/(s-1). (Key point: Y)

2. A=
   A. 1/2
   B. 1/3
   C. -1/6
   D. 1/6
   Answer: D. 1/(2*3). (Key point: A)

3. B=
   A. -1/2
   B. 1/2
   C. 1/3
   D. -1/6
   Answer: A. 1/((-2)(1)). (Key point: B)

4. Solution:
   A. e^t/6+e^(-t)/2+e^(-2t)/3
   B. e^t/6 - e^(-t)/2 + e^(-2t)/3
   C. e^t/6-e^(-t)/2-e^(-2t)/3
   D. e^t/3-e^(-t)/2+e^(-2t)/6
   Answer: B. Inverse. (Key point: y)

## Close variants set in other exams

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