---
title: "Solve ODE by Laplace transform y''+4y'+3y=0"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:solve-ode-by-laplace-transform-y-eight"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/solve-ode-by-laplace-transform-y-eight
site: BCT Engineering
---

# Solve ODE by Laplace transform y''+4y'+3y=0

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2082 Baishakh, 5 marks
- Video: [Linear differential equation using Laplace transform example(PART-3) by easy maths easy tricks](https://www.youtube.com/watch?v=IE1DKTsJL3A) — EASY MATHS EASY TRICKS
- 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-eight
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

By using Laplace transform, solve the initial value problem y'' + 4y' + 3y = 0, given that y(0) = y'(0) = 1.

## Short answer

Transform yields (s²+4s+3)Y = s+5, so Y = 2/(s+1)−1/(s+3). Inverse Laplace gives y(t)=2e^(−t)−e^(−3t).

## Answer

Transform y'' + 4y' + 3y = 0 with y(0) = 1, y'(0) = 1. Let Y = L{y}.

```text
(s²Y − s − 1) + 4(sY − 1) + 3Y = 0
(s² + 4s + 3) Y = s + 5
Y = (s + 5)/[(s + 1)(s + 3)]
```

Partial fractions: Y = A/(s + 1) + B/(s + 3).

```text
A = (−1 + 5)/(−1 + 3) = 2
B = (−3 + 5)/(−3 + 1) = −1
Y = 2/(s + 1) − 1/(s + 3)
```

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

## Check yourself

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

1. y'' + 4y' + 3y = 0, y(0) = y'(0) = 1. What is (s^2+4s+3)Y?
   A. s + 3
   B. s + 1
   C. s - 5
   D. s + 5
   Answer: D. s^2Y - s - 1 + 4sY - 4 + 3Y = 0 gives (s^2+4s+3)Y = s + 5. (Key point: Right side after the transform)

2. Y = (s+5)/((s+1)(s+3)) = A/(s+1) + B/(s+3). What is A?
   A. 2
   B. 4
   C. 1
   D. -2
   Answer: A. A = (-1+5)/(-1+3) = 4/2 = 2. (Key point: Finding A)

3. What is y(t)?
   A. 2e^(t) - e^(3t)
   B. 2e^(-t) - e^(-3t)
   C. e^(-t) + e^(-3t)
   D. 2e^(-t) + e^(-3t)
   Answer: B. Y = 2/(s+1) - 1/(s+3); y(0) = 1 and y'(0) = -2 + 3 = 1 check. (Key point: Final solution)

## Close variants set in other exams

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