---
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-four"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/solve-ode-by-laplace-transform-y-four
site: BCT Engineering
---

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

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2078 Kartik, 5 marks; 2073 Chaitra, 5 marks (asked 2 times)
- 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-four
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Solve the initial value problem y'' + 4y' + 3y = 0, y(0) = 3, y'(0) = 1 by using Laplace transform.

## Short answer

Laplace transform leads to Y = (3s+13)/[(s+1)(s+3)]. Partial fractions give 5/(s+1) - 2/(s+3); inverting yields y(t) = 5e^(-t) - 2e^(-3t).

## Answer

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

```text
L{y''} = s²Y − 3s − 1
L{y'}  = sY − 3

(s²Y − 3s − 1) + 4(sY − 3) + 3Y = 0
(s² + 4s + 3) Y = 3s + 13
Y = (3s + 13)/[(s + 1)(s + 3)]
```

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

```text
A = (3(−1) + 13)/(−1 + 3) = 10/2 = 5
B = (3(−3) + 13)/(−3 + 1) = 4/(−2) = −2
Y = 5/(s + 1) − 2/(s + 3)
```

**Answer: y(t) = 5e^(−t) − 2e^(−3t).** Check: y(0) = 3 ✓; y'(0) = −5 + 6 = 1 ✓.

## Check yourself

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

1. L{y''}=
   A. s^2Y-1
   B. s^2Y-3s
   C. s^2Y - 3s - 1
   D. s^2Y+3s+1
   Answer: C. s^2Y - s y(0) - y'(0). (Key point: L{y''})

2. Y=
   A. (3s+1)/[(s+1)(s+3)]
   B. (3s+13)/(s^2+4s)
   C. (s+13)/[(s+1)(s+3)]
   D. (3s+13)/[(s+1)(s+3)]
   Answer: D. 3s+1+12. (Key point: Y)

3. A,B=
   A. 5 and -2
   B. 2 and 5
   C. -5 and 2
   D. 3 and 1
   Answer: A. Cover-up. (Key point: Coefficients)

4. Solution:
   A. 5e^t-2e^(3t)
   B. 5e^(-t) - 2e^(-3t)
   C. 2e^(-t)+5e^(-3t)
   D. 3e^(-t)+e^(-3t)
   Answer: B. Inverse. (Key point: y)

## Close variants set in other exams

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