---
title: "Solve ODE by Laplace transform y''+y=sin3t"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:solve-ode-by-laplace-transform-y-twelve"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/solve-ode-by-laplace-transform-y-twelve
site: BCT Engineering
---

# Solve ODE by Laplace transform y''+y=sin3t

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2075 Chaitra, 5 marks
- Video: [Application of Laplace Transformation in Differential equations](https://www.youtube.com/watch?v=THqzPHS8cDE) — Anshuman Purohit
- 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-twelve
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Solve y'' + y = sin 3t, y(0) = 0 by using Laplace transform.

## Short answer

Assuming y'(0)=0, Laplace transform gives (s²+1)Y = 3/(s²+9). Partial fractions Y = (3/8)[1/(s²+1) − 1/(s²+9)]; inverse yields y(t) = (3 sin t − sin 3t)/8.

## Answer

The initial condition y′(0) is missing in the question (scan problem). The evident version is y″ + y = sin 3t with y(0) = 0 and y′(0) = 0, and it is solved here. (If y′(0) = k, add the term k sin t to the answer.)

**Step 1: Take the Laplace transform.** Let Y(s) = L[y].

```text
L[y″] = s²Y − s·y(0) − y′(0) = s²Y
s²Y + Y = 3/(s² + 9)
```

**Step 2: Solve for Y.**

```text
Y = 3 / [(s² + 1)(s² + 9)]
```

Partial fractions: 3/[(s²+1)(s²+9)] = (3/8)[1/(s²+1) − 1/(s²+9)].

**Step 3: Inverse transform.**

```text
y(t) = (3/8) sin t − (3/8)(1/3) sin 3t
     = (3/8) sin t − (1/8) sin 3t
```

**Check:** y(0) = 0 and y′(0) = 3/8 − 3/8 = 0. Also y″ + y = −(3/8) sin t + (9/8) sin 3t + (3/8) sin t − (1/8) sin 3t = sin 3t. ✓

**Answer: y(t) = (3 sin t − sin 3t)/8**

## Check yourself

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

1. What is Y?
   A. 3/(s^2+9)
   B. 1/[(s^2+1)(s^2+9)]
   C. 3/[(s^2+1)(s^2+9)]
   D. 3s/[(s^2+1)(s^2+9)]
   Answer: C. s^2Y + Y = 3/(s^2+9). (Key point: Y(s))

2. What is y(t)?
   A. (3 sin t + sin 3t)/8
   B. (sin t - 3 sin 3t)/8
   C. 3 sin t - sin 3t
   D. (3 sin t - sin 3t)/8
   Answer: D. (3/8) sin t - (3/8)(1/3) sin 3t. (Key point: Final answer)

3. If y'(0) = k, what is added to the answer?
   A. k sin t
   B. k cos t
   C. k sin 3t
   D. k t
   Answer: A. k appears as k/(s^2+1), inverting to k sin t. (Key point: General y'(0) = k)

## Close variants set in other exams

- [Solve ODE by Laplace transform y''+4y=sint](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-16) — Mathematics III, chapter 3, asked 2×

## Related in Mathematics III

- [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 (D²−3D+2)y=e^(3t)underth…](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-d2) — Mathematics III, chapter 3
- [Solve ODE by Laplace transform x''+2x'+5x=e^(−t)sint](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-x) — Mathematics III, chapter 3, asked 17×
- [Solve ODE by Laplace transform y''+2y'+2y=5sinx](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-17) — Mathematics III, chapter 3, asked 17×
- [Solve ODE by Laplace transform y''+2y'−3y=sint](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y) — Mathematics III, chapter 3, asked 14×
- [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×