---
title: "Define power series; solve y'+2xy=0"
subject: "Mathematics"
chapter: 4
key: "semester_2/mathematics:define-power-series-solve-y"
source: https://www.bctengineer.com/q/semester_2/mathematics/define-power-series-solve-y
site: BCT Engineering
---

# Define power series; solve y'+2xy=0

Mathematics · Chapter 4: Solution of Differential Equations in Series and Special Functions

- Asked in: 2069 Bhadra, 5 marks
- Video: [Power Series Solution for ODE: y' - 2xy = 0](https://www.youtube.com/watch?v=q9xNKWU8xJ0) — 3LM Engineering
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/define-power-series-solve-y
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-4

## Question

Define power series. Solve by power series method of differential equation, y' + 2xy = 0.

## Short answer

A power series is an infinite sum Σ a_n (x-x_0)^n. Substituting y = Σ a_n x^n into y'+2xy=0 yields a_1=0 and a_{n+1} = -2 a_{n-1}/(n+1), giving y = a_0 e^{-x^2}.

## Answer

#### Power series
An infinite series of the form

Σ aₙ(x − x₀)ⁿ = a₀ + a₁(x − x₀) + a₂(x − x₀)² + …

(with constant coefficients aₙ) is called a power series about the point x₀. About x₀ = 0 it is Σ aₙxⁿ. It converges for |x − x₀| < R, where R is the radius of convergence.

#### Solve y′ + 2xy = 0
x = 0 is an ordinary point. Let y = Σ aₙxⁿ, so y′ = Σ (n+1)aₙ₊₁ xⁿ and 2xy = Σ 2aₙ₋₁ xⁿ (n ≥ 1).

Coefficient of x⁰: a₁ = 0.
Coefficient of xⁿ (n ≥ 1): (n+1)aₙ₊₁ + 2aₙ₋₁ = 0, so

**aₙ₊₁ = − 2aₙ₋₁ / (n + 1)**

```text
a₁ = 0  →  all odd coefficients a₃ = a₅ = … = 0
a₂ = −2a₀/2 = −a₀
a₄ = −2a₂/4 = a₀/2
a₆ = −2a₄/6 = −a₀/6
```
Pattern: a₂ₘ = (−1)^m a₀ / m!.

```text
y = a₀(1 − x² + x⁴/2! − x⁶/3! + …) = a₀ e^(−x²)
```
**Answer: y = a₀e^(−x²).** (Check by separation of variables: dy/y = −2x dx gives y = Ce^(−x²).)

## Check yourself

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

1. A power series about x0 is an infinite series of the form:
   A. Σ aₙ xⁿ/n! only, about the origin
   B. Σ (x − x0)ⁿ with aₙ depending on x
   C. Σ aₙ(x − x0)ⁿ with constant coefficients aₙ
   D. A finite polynomial in (x − x0)
   Answer: C. It is a sum of powers of (x − x0) with constant coefficients; about 0 it is Σ aₙxⁿ. (Key point: Definition)

2. Why does the coefficient of x^0 give a1 = 0 in y' + 2xy = 0?
   A. 2xy contributes a1 at x^0
   B. y' contributes a1 at x^0, while 2xy has no x^0 term
   C. Because a0 = 0
   D. Because x = 0 is a singular point
   Answer: B. The x^0 term of y' is a1 and 2xy starts at x^1, so a1 = 0. (Key point: Why a1 = 0)

3. What is the recurrence relation obtained?
   A. a(n+1) = −2a(n−1)/(n+1)
   B. a(n+1) = 2a(n−1)/(n+1)
   C. a(n+1) = −2a(n)/(n+1)
   D. a(n+2) = −2a(n)/(n+2)
   Answer: A. Equating the coefficient of xⁿ gives (n+1)a(n+1) + 2a(n−1) = 0. (Key point: Recurrence)

4. The series 1 − x^2 + x^4/2! − x^6/3! + … sums to:
   A. e^(x^2)
   B. cos x^2
   C. 1/(1 + x^2)
   D. e^(−x^2)
   Answer: D. a(2m) = (−1)^m a0/m!, which is the series of e^(−x^2); it matches separation of variables. (Key point: Final solution)

## Close variants set in other exams

- [Power series: y''−4xy'+(4x²−2)y=0](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Apower-series-y-4xy-4x2) — Mathematics, chapter 4, asked 12×
- [Power series solution of y''+xy'+y=0](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Apower-series-solution-y-xy) — Mathematics, chapter 4, asked 12×
- [Power series solution of y''−y=0](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Apower-series-solution-y-y) — Mathematics, chapter 4, asked 12×
- [Power series solution of y''+x²y=0](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Apower-series-solution-y-x2y) — Mathematics, chapter 4, asked 12×
- [Power series solution of (1−x²)y''−2xy'+2y=0](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Apower-series-solution-1-x2) — Mathematics, chapter 4, asked 12×

## Related in Mathematics

- [Series solution of Legendre's equation](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aseries-solution-legendre-s-equation) — Mathematics, chapter 4, asked 4×
- [Power series solution of y''−y=x](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Apower-series-solution-y-y-2) — Mathematics, chapter 4

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