---
title: "Power series solution of y''−y=x"
subject: "Mathematics"
chapter: 4
key: "semester_2/mathematics:power-series-solution-y-y-2"
source: https://www.bctengineer.com/q/semester_2/mathematics/power-series-solution-y-y-2
site: BCT Engineering
---

# Power series solution of y''−y=x

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

- Asked in: 2073 Magh, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/power-series-solution-y-y-2
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-4

## Question

Solve by power series method y'' − y = x.

## Short answer

Substituting y = Σ a_n x^n gives recurrence a_{n+2} = a_n/[(n+1)(n+2)] for n≠1, with 6a_3 - a_1 = 1. This yields y = a_0 cosh x + (a_1+1) sinh x - x, i.e., y = C_1 e^x + C_2 e^{-x} - x.

## Answer

Equation: y″ − y = x. Take y = Σ aₙxⁿ, so y″ = Σ (n+2)(n+1)aₙ₊₂ xⁿ.

#### Equate coefficients of xⁿ
```text
(n+2)(n+1)aₙ₊₂ − aₙ = 0     for n ≠ 1
 6a₃ − a₁ = 1               for n = 1  (coefficient of x on the right is 1)
```
So aₙ₊₂ = aₙ/((n+1)(n+2)) except a₃ = (a₁ + 1)/6.

#### Coefficients
```text
Even:  a₂ = a₀/2!,  a₄ = a₀/4!,  a₆ = a₀/6! …
Odd:   a₃ = (a₁+1)/3!,  a₅ = a₃/20 = (a₁+1)/5!,  a₇ = (a₁+1)/7! …
```

#### Solution
```text
y = a₀(1 + x²/2! + x⁴/4! + …) + a₁x + (a₁+1)(x³/3! + x⁵/5! + …)
  = a₀ cosh x + (a₁+1)(x + x³/3! + x⁵/5! + …) − x
  = a₀ cosh x + (a₁+1) sinh x − x
```
Writing the constants as C₁ and C₂:

**y = C₁eˣ + C₂e⁻ˣ − x**

Check: for y = −x, y″ − y = 0 + x = x ✓ (this is the particular solution).

## Check yourself

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

1. How does RHS x affect the recurrence?
   A. The coefficient equation at n = 1 becomes 6a3 - a1 = 1
   B. It changes all equations
   C. It adds 1 to a0
   D. It removes a1
   Answer: A. Only the x^1 coefficient on the right is nonzero. (Key point: Nonhomogeneous term)

2. a3 equals:
   A. a1/6
   B. (a1 + 1)/6
   C. (a1 - 1)/6
   D. (a1 + 1)/3
   Answer: B. 6a3 = a1 + 1. (Key point: a3)

3. The particular solution is:
   A. y = x
   B. y = x^2
   C. y = -x
   D. y = e^x
   Answer: C. y'' - y = 0 + x. (Key point: Particular solution)

4. The general solution is:
   A. C1 e^x + C2 e^(-x) - x
   B. C1 e^x + C2 e^(-x) + x
   C. C1 cos x + C2 sin x - x
   D. C1 e^x + C2 e^(-x)
   Answer: A. Complementary plus particular. (Key point: General solution)

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