---
title: "String with initial deflection lx - x²"
subject: "Applied Mathematics"
chapter: 3
key: "semester_4/applied_mathematics:wave-eq-string-lx-x2"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/wave-eq-string-lx-x2
site: BCT Engineering
---

# String with initial deflection lx - x²

Applied Mathematics · Chapter 3: Partial Differential Equations

- Asked in: 2071 Bhadra
- Video: [Problem on wave Equation](https://www.youtube.com/watch?v=TyLBj7yO-LY) — Sreeja's MathWorld
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/wave-eq-string-lx-x2
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-3

## Question

Solve the wave equation for a tightly stretched string of length l fixed at both ends if the initial deflection is y(x, 0) = lx - x² and the initial velocity is zero.

## Short answer

With fixed ends and zero initial velocity, the solution is a sine series in x with cosine time dependence. Fourier coefficients are Bₙ = (2/l)∫₀^l (lx−x²) sin(nπx/l)dx, non-zero only for odd n. y(x,t) = (8l²/π³) Σ_{n odd} (1/n³) cos(nπct/l) sin(nπx/l).

## Answer

**Problem.** y_tt = c² y_xx on 0 < x < l, with y(0, t) = y(l, t) = 0, y(x, 0) = lx − x², and zero initial velocity.

**Step 1: general solution** (separation of variables with fixed ends):

y(x, t) = Σ_{n=1}^{∞} [ Bₙ cos(nπct/l) + Bₙ* sin(nπct/l) ] sin(nπx/l)

**Step 2: zero initial velocity.** y_t(x, 0) = Σ Bₙ* (nπc/l) sin(nπx/l) = 0, so Bₙ* = 0.

**Step 3: initial deflection.** y(x, 0) = lx − x² = Σ Bₙ sin(nπx/l), so

Bₙ = (2/l) ∫₀^l (lx − x²) sin(nπx/l) dx

Integrating by parts twice gives ∫₀^l (lx − x²) sin(nπx/l) dx = 2l³[1 − (−1)ⁿ]/(n³π³). Hence

```text
Bₙ = 4l²[1 − (−1)ⁿ]/(n³π³)
   = 8l²/(n³π³)   for n odd
   = 0            for n even
```

**Answer:**

y(x, t) = (8l²/π³) Σ_{n = 1,3,5,…} (1/n³) cos(nπct/l) sin(nπx/l)

Check: at t = 0 and l = 2, x = 0.7, the series sums to 0.9100 = lx − x² ✓ (computed numerically).

## Check yourself

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

1. Zero initial velocity gives
   A. Bn* = 0
   B. Bn = 0
   C. Bn* = Bn
   D. c = 0
   Answer: A. No sine in t. (Key point: Velocity)

2. Bn =
   A. (1/l) integral
   B. (2/l) integral of (lx - x^2) sin(n pi x/l) dx
   C. (2/l) integral with cos
   D. integral over 0 to 2l
   Answer: B. Fourier sine. (Key point: Coefficient)

3. Bn is nonzero for
   A. even n only
   B. all n
   C. odd n only
   D. n = 1 only
   Answer: C. 1-(-1)^n. (Key point: Parity)

4. y =
   A. (8 l^2/pi^3) sum over odd n of sin(n pi c t/l) sin(n pi x/l)/n^3
   B. (8 l^2/pi^3) sum over odd n of cos(n pi c t/l) sin(n pi x/l)/n
   C. (24 l^3/(c pi^4)) sum over odd n of sin(n pi c t/l) sin(n pi x/l)/n^4
   D. (8 l^2/pi^3) sum over odd n of cos(n pi c t/l) sin(n pi x/l)/n^3
   Answer: D. Bn = 8 l^2/(n pi)^3. (Key point: Result)

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