---
title: "Solve ODE by Laplace transform y''−10y'+9y=5t"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:solve-ode-by-laplace-transform-y-eleven"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/solve-ode-by-laplace-transform-y-eleven
site: BCT Engineering
---

# Solve ODE by Laplace transform y''−10y'+9y=5t

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2078 Bhadra, 5 marks
- Video: [Solve initial value problem y"+2y'+2y=0, y(0)=1, y'(0) =-3 using Laplace Transform Engineering math](https://www.youtube.com/watch?v=l6uJ1JoteRY) — Swati Theng Mathematics
- 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-eleven
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Solve the initial value problem by applying the Laplace transform: y'' − 10y' + 9y = 5t, y(0) = −1, y'(0) = 2.

## Short answer

Apply Laplace transform and partial fractions; solution: y(t) = 50/81 + 5t/9 - 2e^t + (31/81) e^{9t}.

## Answer

Transform y'' − 10y' + 9y = 5t with y(0) = −1, y'(0) = 2. Let Y = L{y}.

```text
(s²Y + s − 2) − 10(sY + 1) + 9Y = 5/s²
(s² − 10s + 9) Y = −s + 12 + 5/s²
Y = (12 − s)/[(s − 1)(s − 9)] + 5/[s²(s − 1)(s − 9)]
```

Partial fractions:

```text
(12 − s)/[(s − 1)(s − 9)] = (−11/8)/(s − 1) + (3/8)/(s − 9)
5/[s²(s − 1)(s − 9)]      = (50/81)/s + (5/9)/s² − (5/8)/(s − 1) + (5/648)/(s − 9)

Y = (50/81)/s + (5/9)/s² − 2/(s − 1) + (31/81)/(s − 9)
```

Inverse transform:

**Answer: y(t) = 50/81 + 5t/9 − 2eᵗ + (31/81) e^(9t).** Check: y(0) = 50/81 − 2 + 31/81 = −1 ✓; y'(0) = 5/9 − 2 + 31/9 = 2 ✓.

## Check yourself

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

1. With y(0) = -1, y'(0) = 2, what is L{y''}?
   A. s^2 Y + s - 2
   B. s^2 Y - s - 2
   C. s^2 Y - s + 2
   D. s^2 Y + s + 2
   Answer: A. s^2Y - s(-1) - 2 = s^2Y + s - 2. (Key point: Transform of y'')

2. s^2 - 10s + 9 factors as?
   A. (s+1)(s+9)
   B. (s-1)(s-9)
   C. (s-3)(s-3)
   D. (s-1)(s+9)
   Answer: B. Roots 1 and 9 (sum 10, product 9). (Key point: Factoring)

3. The two parts give -11/8 and -5/8 on 1/(s-1). What is the total e^t coefficient?
   A. -11/8
   B. -5/8
   C. -2
   D. -3/2
   Answer: C. -11/8 - 5/8 = -16/8 = -2. (Key point: Coefficient of e^t)

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