---
title: "Power series: y''−4xy'+(4x²−2)y=0"
subject: "Mathematics"
chapter: 4
key: "semester_2/mathematics:power-series-y-4xy-4x2"
source: https://www.bctengineer.com/q/semester_2/mathematics/power-series-y-4xy-4x2
site: BCT Engineering
---

# Power series: y''−4xy'+(4x²−2)y=0

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

- Asked in: 2073 Bhadra, 5 marks; 2072 Ashwin, 5 marks; 2070 Bhadra, 5 marks; 2068 Bhadra, 5 marks (asked 4 times)
- Video: [y" -4xy' +(4x^2 -2)y = 0 Power Series of Differential Equation](https://www.youtube.com/watch?v=bpKeEZ2zV00) — Confused Student
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/power-series-y-4xy-4x2
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-4

## Question

Solve by power series method the differential equation y'' − 4xy' + (4x²−2)y = 0.

## Short answer

Assume y = Σ a_n x^n. The recurrence gives a_{2k}=a_0/k! and a_{2k+1}=a_1/k!, so y = a_0 e^(x²) + a_1 x e^(x²) = (a_0 + a_1 x) e^(x²).

## Answer

Equation: y″ − 4xy′ + (4x² − 2)y = 0. x = 0 is an ordinary point, so let y = Σ aₙxⁿ.

#### Substitution
y″ = Σ (n+2)(n+1) aₙ₊₂ xⁿ, −4xy′ = −Σ 4n aₙ xⁿ, 4x²y = Σ 4aₙ₋₂ xⁿ, −2y = −Σ 2aₙ xⁿ.

Coefficient of xⁿ:

```text
(n+2)(n+1) aₙ₊₂ − 4n aₙ − 2aₙ + 4aₙ₋₂ = 0
aₙ₊₂ = [(4n + 2) aₙ − 4 aₙ₋₂] / ((n+2)(n+1))
```
(with a₋₁ = a₋₂ = 0).

#### Coefficients
```text
n=0:  2a₂ = 2a₀                 → a₂ = a₀
n=1:  6a₃ = 6a₁                 → a₃ = a₁
n=2:  12a₄ = 10a₂ − 4a₀ = 6a₀   → a₄ = a₀/2
n=3:  20a₅ = 14a₃ − 4a₁ = 10a₁  → a₅ = a₁/2
n=4:  30a₆ = 18a₄ − 4a₂ = 5a₀   → a₆ = a₀/6
```
a₀ and a₁ are arbitrary.

#### Solution
```text
y = a₀(1 + x² + x⁴/2! + x⁶/3! + …) + a₁(x + x³ + x⁵/2! + x⁷/3! + …)
```
The first series is e^(x²), the second is x·e^(x²).

**Answer: y = (a₀ + a₁x) e^(x²).**

Check by substitution: for y = e^(x²), y″ = (2 + 4x²)e^(x²), y′ = 2x e^(x²), so y″ − 4xy′ + (4x² − 2)y = (2 + 4x² − 8x² + 4x² − 2)e^(x²) = 0.

## Check yourself

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

1. The recurrence is:
   A. a_(n+2) = [(4n-2) a_n + 4 a_(n-2)] / ((n+2)(n+1))
   B. a_(n+2) = (4n+2) a_n / ((n+2)(n+1))
   C. a_(n+2) = [(4n+2) a_n - 4 a_(n-2)] / ((n+2)(n+1))
   D. a_(n+1) = (4n+2) a_n - 4 a_(n-2)
   Answer: C. Collect x^n terms: three indices a_(n+2), a_n, a_(n-2). (Key point: Recurrence)

2. The first coefficients are a2 = a0 and a3 = a1. What is a4?
   A. a0/2
   B. a0
   C. a0/4
   D. a0/6
   Answer: A. 12 a4 = 10 a2 - 4 a0 = 6 a0. (Key point: Coefficients)

3. The even series 1 + x^2 + x^4/2! + x^6/3! + ... is:
   A. e^x
   B. e^(x^2)
   C. cosh x
   D. e^(-x^2)
   Answer: B. It is sum (x^2)^m / m!. (Key point: Even series)

4. The general solution is:
   A. y = (a0 + a1 x) e^(-x^2)
   B. y = a0 e^(x^2) + a1 e^(-x^2)
   C. y = a0 + a1 x
   D. y = (a0 + a1 x) e^(x^2)
   Answer: D. The odd series is x e^(x^2). (Key point: Solution)

## Close variants set in other exams

- [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×
- [Define power series; solve y'+2xy=0](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Adefine-power-series-solve-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
- [Convergence of 2x+3x²/8+4x³/27+...](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aconvergence-2x-3x2-8-4x3) — Mathematics, chapter 6, asked 3×
- [Series solution of x²y''+xy'+(x²−4)y=0](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aseries-solution-x2y-xy-x2) — Mathematics, chapter 4

## Related in other subjects

- [Solve a partial differential equation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-a-partial-differential-equation) — Mathematics III, chapter 4
- [Solve a partial differential equation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-a-partial-differential-equation-two) — Mathematics III, chapter 4
- [Solve x²y''+4xy'+2y=eˣ](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-cauchy-x2y2-plus-4xy1-plus-2y) — Mathematics I, chapter 4, asked 2×
- [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×