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

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

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2075 Asoj, 5 marks; 2071 Shrawan, 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-five
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Solve the following initial value problem by using Laplace transform: y'' + 4y' + 3y = eᵗ, y(0) = 0, y'(0) = 2.

## Short answer

Transform gives Y = (2s-1)/[(s-1)(s+1)(s+3)]. Partial fractions: A=1/8, B=3/4, C=-7/8. Inverse: y(t) = e^t/8 + (3/4)e^(-t) - (7/8)e^(-3t).

## Answer

Transform y'' + 4y' + 3y = eᵗ with y(0) = 0, y'(0) = 2. Let Y = L{y}.

```text
(s²Y − 0·s − 2) + 4(sY − 0) + 3Y = 1/(s − 1)
(s² + 4s + 3) Y = 2 + 1/(s − 1) = (2s − 1)/(s − 1)
Y = (2s − 1)/[(s − 1)(s + 1)(s + 3)]
```

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

```text
A = (2 − 1)/[(2)(4)] = 1/8
B = (−2 − 1)/[(−2)(2)] = 3/4
C = (−6 − 1)/[(−4)(−2)] = −7/8
```

**Answer: y(t) = eᵗ/8 + (3/4) e^(−t) − (7/8) e^(−3t).** Check: y(0) = 1/8 + 6/8 − 7/8 = 0 ✓; y'(0) = 1/8 − 3/4 + 21/8 = 2 ✓.

## Check yourself

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

1. (s^2+4s+3)Y=
   A. 1/(s-1)
   B. 2-1/(s-1)
   C. 2 + 1/(s-1)
   D. 2s+1/(s-1)
   Answer: C. y'(0)=2 adds 2. (Key point: RHS)

2. Y=
   A. 1/[(s-1)(s+1)(s+3)]
   B. (2s+1)/[...]
   C. (2s-1)/[(s+1)(s+3)]
   D. (2s-1)/[(s-1)(s+1)(s+3)]
   Answer: D. Combine. (Key point: Y)

3. A=
   A. 1/8
   B. 1/4
   C. 3/4
   D. -7/8
   Answer: A. (2-1)/(2*4). (Key point: A)

4. Solution:
   A. e^t/8 - 3e^(-t)/4 + 7e^(-3t)/8
   B. e^t/8 + 3e^(-t)/4 - 7e^(-3t)/8
   C. e^t/8+3e^(-t)/4+7e^(-3t)/8
   D. e^t/4+e^(-t)/2-e^(-3t)
   Answer: B. Inverse. (Key point: y)

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