---
title: "Express 5x³+x in Legendre polynomials"
subject: "Mathematics"
chapter: 4
key: "semester_2/mathematics:express-5x3-x-legendre-polynomials"
source: https://www.bctengineer.com/q/semester_2/mathematics/express-5x3-x-legendre-polynomials
site: BCT Engineering
---

# Express 5x³+x in Legendre polynomials

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

- Asked in: 2074 Bhadra, 5 marks
- Video: [Legendre's Equation | Legendre Polynomials | Numericals | Series solution of ODEs | Maths](https://www.youtube.com/watch?v=BA4I6OhOmFI) — Gautam Varde
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/express-5x3-x-legendre-polynomials
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-4

## Question

Express the following in terms of Legendre's Polynomials: f(x) = 5x³ + x.

## Short answer

From P_1 = x and P_3 = (5x^3 - 3x)/2, solve for 5x^3 = 2P_3 + 3P_1. Then f(x) = 5x^3 + x = 2P_3 + 4P_1.

## Answer

Legendre polynomials: P₁ = x, P₃ = (5x³ − 3x)/2.

#### Express x³
From P₃ = (5x³ − 3x)/2 we get 5x³ = 2P₃ + 3x = 2P₃ + 3P₁.

#### Substitute
f(x) = 5x³ + x = (2P₃ + 3P₁) + P₁ = 2P₃ + 4P₁.

**Answer: 5x³ + x = 2P₃(x) + 4P₁(x).**

Check: 2(5x³ − 3x)/2 + 4x = 5x³ − 3x + 4x = 5x³ + x ✓.

(Alternatively, put f = c₀P₀ + c₁P₁ + c₂P₂ + c₃P₃ and compare coefficients: x³ gives c₃ = 2, x gives c₁ − 3 = 1 so c₁ = 4, and c₂ = c₀ = 0.)

## Check yourself

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

1. 5x^3 equals:
   A. 2P3 + 3P1
   B. 2P3 - 3P1
   C. P3 + 3P1
   D. 2P3 + P1
   Answer: A. From P3 = (5x^3-3x)/2. (Key point: Express x^3)

2. Adding x = P1 to 5x^3 gives:
   A. 2P3 + 3P1
   B. 2P3 + 4P1
   C. 2P3 + 2P1
   D. P3 + 4P1
   Answer: B. 3P1 + P1. (Key point: Substitute)

3. The coefficients of P0 and P2 are:
   A. Both 1
   B. P0 is 1, P2 is 0
   C. Both zero
   D. Both nonzero
   Answer: C. f is odd, so only odd P_n appear. (Key point: Check)

## Close variants set in other exams

- [Express x³−5x²+6x+1 in Legendre polynomials](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aexpress-x3-5x2-6x-1) — Mathematics, chapter 4, asked 3×

## Related in Mathematics

- [Express x³−5x²+x+2 in Legendre polynomials](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aexpress-x3-5x2-x-2) — Mathematics, chapter 4, asked 3×