---
title: "Solve ODE by Laplace transform y''−3y'+2y=4x+e^(3x)"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:solve-ode-by-laplace-transform-y-ten"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/solve-ode-by-laplace-transform-y-ten
site: BCT Engineering
---

# Solve ODE by Laplace transform y''−3y'+2y=4x+e^(3x)

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2080 Baishakh, 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-y-ten
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Solve the initial value problem by the Laplace transform method: y'' − 3y' + 2y = 4x + e^(3x), y(0) = 1, y'(0) = −1.

## Short answer

y(x) = 3 + 2x - e^x/2 - 2 e^(2x) + e^(3x)/2.

## Answer

The independent variable is x; use t in the transform. Solve y'' − 3y' + 2y = 4x + e^(3x), y(0) = 1, y'(0) = −1. Let Y = L{y}.

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

Partial fractions of each part:

```text
(s − 4)/[(s − 1)(s − 2)]      = 3/(s − 1) − 2/(s − 2)
4/[s²(s − 1)(s − 2)]          = 3/s + 2/s² − 4/(s − 1) + 1/(s − 2)
1/[(s − 1)(s − 2)(s − 3)]     = (1/2)/(s − 1) − 1/(s − 2) + (1/2)/(s − 3)
```

Adding:

```text
Y = 3/s + 2/s² − (1/2)/(s − 1) − 2/(s − 2) + (1/2)/(s − 3)
```

Inverse transform:

**Answer: y(x) = 3 + 2x − eˣ/2 − 2e^(2x) + e^(3x)/2.** Check: y(0) = 3 − 1/2 − 2 + 1/2 = 1 ✓; y'(0) = 2 − 1/2 − 4 + 3/2 = −1 ✓.

## Check yourself

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

1. The independent variable is x. How is Laplace used?
   A. Treat x as t, transform, then rename t back to x
   B. Laplace cannot be used
   C. Transform with respect to y
   D. Differentiate the equation first
   Answer: A. The transform is variable-name independent; write t, solve, then rewrite the answer in x. (Key point: Variable is x)

2. y(0) = 1, y'(0) = -1. (s^2-3s+2)Y = ? (before the forcing terms)
   A. s - 2
   B. s - 4
   C. s + 4
   D. s + 2
   Answer: B. s^2Y - s + 1 - 3sY + 3 + 2Y gives s - 1 - 3 = s - 4. (Key point: Constant on the right)

3. In 1/[(s-1)(s-2)(s-3)], the coefficient of 1/(s-3) is?
   A. 1
   B. 1/6
   C. 1/2
   D. -1/2
   Answer: C. Put s = 3: 1/[(2)(1)] = 1/2. (Key point: e^(3x) coefficient)

4. What is y(x)?
   A. 3 + 2x + e^x/2 - 2e^(2x) + e^(3x)/2
   B. 3 + 2x - e^x/2 + 2e^(2x) - e^(3x)/2
   C. 2x - e^x/2 - 2e^(2x) + e^(3x)/2
   D. 3 + 2x - e^x/2 - 2e^(2x) + e^(3x)/2
   Answer: D. Summing the three partial-fraction parts and inverting gives this; y(0) = 1 and y'(0) = -1 check. (Key point: Final answer)

## Close variants set in other exams

- [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''+3y'+2y=e^(−t)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-two) — Mathematics III, chapter 3, asked 18×
- [Solve ODE by Laplace transform y''+4y'+3y=0](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-four) — Mathematics III, chapter 3, asked 18×
- [Solve ODE by Laplace transform y''+4y'+3y=eᵗ](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-five) — Mathematics III, chapter 3, asked 16×
- [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''+4y'+3y=0](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-eight) — Mathematics III, chapter 3, asked 13×
- [Solve ODE by Laplace transform y''−10y'+9y=5t](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-eleven) — Mathematics III, chapter 3, asked 13×
- [Solve ODE by Laplace transform x''+4x'+4x=6e^(−t)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-x-two) — Mathematics III, chapter 3, asked 16×
- [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×

## Related in Mathematics III

- [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''+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=r(t)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-13) — Mathematics III, chapter 3, asked 5×
- [Solve ODE by Laplace transform y''+4y'+3y=t](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-14) — Mathematics III, chapter 3
- [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'''+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×

## Related in other subjects

- [Initial value problem y''−4y'+3y](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Ainitial-value-problem-y-4y) — Mathematics, chapter 4, asked 3×
- [Initial value problem y''+2y'+5y=0](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Ainitial-value-problem-y-2y) — Mathematics, chapter 4, asked 3×
- [Solve y''+3y'+2y=e^(2x) sin x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-d2-3d-2-e2x-sinx) — Mathematics I, chapter 4, asked 2×
- [Solve y''−2y'+y=x²e^(3x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-y2-minus-2y1-plus-y-x2e3x) — Mathematics I, chapter 4