---
title: "Solve ODE by Laplace transform (D²−3D+2)y=e^(3t)underth…"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:solve-ode-by-laplace-transform-d2"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/solve-ode-by-laplace-transform-d2
site: BCT Engineering
---

# Solve ODE by Laplace transform (D²−3D+2)y=e^(3t)underth…

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2081 Bhadra, 5 marks
- Video: [Question #4 - Laplace Transform - Application to Ordinary Differential Equation.](https://www.youtube.com/watch?v=1ovNMSQ3PVs) — Dr. Chand Ram
- 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-d2
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Solve the differential equation (D² − 3D + 2)y = e^(3t) under the conditions y(0) = 0, y'(0) = 1, using the Laplace transform.

## Short answer

y(t) = (e^(3t) - e^t)/2.

## Answer

The equation (D² − 3D + 2)y = e^(3t) means y'' − 3y' + 2y = e^(3t), with y(0) = 0, y'(0) = 1. Let Y = L{y}.

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

Inverse transform:

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

## Check yourself

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

1. y(0) = 0, y'(0) = 1. What is (s^2 - 3s + 2)Y?
   A. 1/(s-3)
   B. s + 1/(s-3)
   C. 1 + 1/(s-3) = (s-2)/(s-3)
   D. 1 - 1/(s-3)
   Answer: C. The initial condition y'(0) = 1 moves over as +1, and L{e^(3t)} = 1/(s-3). (Key point: Transformed equation)

2. (s-1)(s-2)Y = (s-2)/(s-3). What does Y become?
   A. (s-2)/((s-1)(s-3))
   B. 1/((s-1)(s-2)(s-3))
   C. 1/((s-2)(s-3))
   D. 1/((s-1)(s-3))
   Answer: D. The factor (s-2) cancels from both sides. (Key point: Simplifying Y)

3. What is y(t)?
   A. (e^(3t) - e^t)/2
   B. (e^t - e^(3t))/2
   C. e^(3t) - e^t
   D. (e^(3t) + e^t)/2
   Answer: A. 1/((s-1)(s-3)) = (1/2)[1/(s-3) - 1/(s-1)]. (Key point: Final solution)

## Related in Mathematics III

- [Solve ODE by Laplace transform y''+y'−2y=x](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-seven) — Mathematics III, chapter 3, asked 5×
- [Solve ODE by Laplace transform y''+3y'+2y=eᵗ](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-three) — Mathematics III, chapter 3, asked 4×
- [Solve ODE by Laplace transform y'''+2y''−y'−2y=0](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-15) — Mathematics III, chapter 3, asked 3×
- [Solve ODE by Laplace transform y''+2y'+5y=e^(−t)sint](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-six) — Mathematics III, chapter 3, asked 2×
- [Solve ODE by Laplace transform y''+y=sin3t](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-nine) — Mathematics III, chapter 3, asked 12×
- [Solve ODE by Laplace transform y''+y=sin3t](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-twelve) — Mathematics III, chapter 3, asked 2×

## Related in other subjects

- [Laplace equation with u(x,∞) = sin(nπx/l)](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Alaplace-eq-u-x-inf-sin) — Applied Mathematics, chapter 3
- [Laplace equation solutions; u(x,0)=sin x](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Alaplace-eq-sin-x-bounded) — Applied Mathematics, chapter 3