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

# String with initial velocity 3(lx-x²)

Applied Mathematics · Chapter 3: Partial Differential Equations

- Asked in: 2075 Baisakh, 10 marks; 2073 Bhadra, 10 marks (asked 2 times)
- Video: [MA25C03 |MA3351 | One Dimensional Wave Equation (Non Zero Initial Velocity )|Vbrating String Problem](https://www.youtube.com/watch?v=ZWKfysqXZHw) — Mathematics Kala
- 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-velocity-3lx-x2
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-3

## Question

A tightly stretched string with fixed ends at x = 0 and x = l is initially at rest in its equilibrium position. Find the deflection u(x, t) if it is set vibrating by giving to each of its points a velocity 3(lx - x²). [Equivalently: solve completely ∂²u/∂t² = c² ∂²u/∂x² under u(0, t) = 0, u(l, t) = 0, u(x, 0) = 0 and (∂u/∂t) at t=0 = 3(lx - x²).]

## Short answer

Separation of variables with u(x,0)=0 yields sine series in t. Matching initial velocity 3(lx-x²) via Fourier sine series gives u(x,t)=(24l³/(cπ⁴))∑_{m=1}∞ sin((2m-1)πct/l) sin((2m-1)πx/l)/(2m-1)⁴.

## Answer

**Problem.** ∂²u/∂t² = c² ∂²u/∂x², with

- u(0, t) = u(l, t) = 0
- u(x, 0) = 0
- ∂u/∂t at t = 0 equals g(x) = 3(lx − x²)

**Step 1: separation of variables.** Put u = X(x)T(t). Then X″ + p²X = 0 and T″ + c²p²T = 0.

The ends being fixed gives X = sin(nπx/l), p = nπ/l (n = 1, 2, …).

T(t) = C cos(nπct/l) + D sin(nπct/l)

**Step 2: u(x, 0) = 0** means every cosine coefficient is zero, so the solution has only sine terms in t:

u(x, t) = Σ_{n=1}^{∞} Bₙ sin(nπct/l) sin(nπx/l)

**Step 3: velocity condition.** Differentiate at t = 0:

```text
∂u/∂t |_{t=0} = Σ Bₙ (nπc/l) sin(nπx/l) = 3(lx − x²)
```

This is a Fourier sine series, so

Bₙ (nπc/l) = (2/l) ∫₀^l 3(lx − x²) sin(nπx/l) dx

**Step 4: the integral.** Integrate by parts twice:

∫₀^l (lx − x²) sin(nπx/l) dx = 2l³[1 − (−1)ⁿ] / (n³π³)

which equals 4l³/(n³π³) for odd n and 0 for even n.

```text
Bₙ (nπc/l) = (2/l) · 3 · 4l³/(n³π³) = 24 l² /(n³π³)   (n odd)
Bₙ = 24 l³ / (c π⁴ n⁴)                                (n odd)
```

**Answer:**

u(x, t) = (24 l³ / (c π⁴)) Σ_{n = 1,3,5,…} (1/n⁴) sin(nπct/l) sin(nπx/l)

or, writing n = 2m − 1: u(x, t) = (24l³/(cπ⁴)) Σ_{m=1}^{∞} sin((2m−1)πct/l) sin((2m−1)πx/l)/(2m−1)⁴.

(Numerical check: l = 2, c = 1.3, x = 0.7: the series for ∂u/∂t at t = 0 sums to 2.7300, equal to 3(lx − x²) = 2.73.)

## Check yourself

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

1. u(x,0)=0 eliminates
   A. the cos(n pi c t/l) terms
   B. the sin terms
   C. the sin(n pi x/l) terms
   D. the velocity
   Answer: A. Leaves sin in t. (Key point: Cosine terms)

2. u_t at t=0 gives
   A. Bn as the coefficient of g(x), with no extra factor
   B. Bn (n pi c/l) as the Fourier sine coefficient of g(x)
   C. Bn l/(n pi c) as the Fourier sine coefficient of g(x)
   D. a cosine series in x
   Answer: B. Differentiating brings n pi c/l. (Key point: Velocity condition)

3. The sine integral of (lx - x^2) 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: Integral)

4. Bn =
   A. 24 l^2/(n^3 pi^3)
   B. 8 l^2/(n^3 pi^3)
   C. 24 l^3/(c pi^4 n^3)
   D. 24 l^3/(c pi^4 n^4) for odd n
   Answer: D. Divide by n pi c/l. (Key point: Result)

## Related in Applied Mathematics

- [String with initial deflection lx - x²](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Awave-eq-string-lx-x2) — Applied Mathematics, chapter 3
- [String released from sin³(πx/l)](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Awave-eq-string-sin3) — Applied Mathematics, chapter 3, asked 2×
- [Heat equation, insulated ends, u(x,0)=lx-x²](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aheat-eq-insulated-ends-lx-x2) — Applied Mathematics, chapter 3, asked 2×

## Related in other subjects

- [Vibrating string deflection](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Avibrating-string-deflection) — Mathematics III, chapter 4
- [Lateral vibration of a stretched string](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Astring-wave-equation-lateral-vibration) — Theory of Mechanism and Machine II, chapter 10