---
title: "Laplace equation solutions; u(x,0)=sin x"
subject: "Applied Mathematics"
chapter: 3
key: "semester_4/applied_mathematics:laplace-eq-sin-x-bounded"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/laplace-eq-sin-x-bounded
site: BCT Engineering
---

# Laplace equation solutions; u(x,0)=sin x

Applied Mathematics · Chapter 3: Partial Differential Equations

- Asked in: 2071 Magh, 10 marks
- Video: [LAPLACE EQUATION WITH EXAMPLE PROBLEM ||M3||OU EDUCATION](https://www.youtube.com/watch?v=UqqZEuSkjN8) — OU Education
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/laplace-eq-sin-x-bounded
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-3

## Question

Find all possible solutions of Laplace equation u_xx + u_yy = 0. Using this, hence solve u_xx + u_yy = 0, under the conditions u(0, y) = 0, u(π, y) = 0 [as printed: u(0,y) = u(x,y) = 0] when y → ∞ and u(x, 0) = sin x.

## Short answer

All solutions are combinations: k=0 gives linear/constant; k>0 gives exponentials in x and trig in y; k<0 gives trig in x and exponentials in y. For given conditions, u(x,y) = e^{−y} sin x.

## Answer

#### Part 1: All possible solutions of u_xx + u_yy = 0

Assume u = X(x)Y(y). Then X″Y + XY″ = 0, so X″/X = −Y″/Y = k (a constant). Three cases arise.

**Case 1: k = 0.** X″ = 0 and Y″ = 0:

u = (c₁x + c₂)(c₃y + c₄)

**Case 2: k = p² > 0.** X″ − p²X = 0 and Y″ + p²Y = 0:

u = (c₅ e^{px} + c₆ e^{−px})(c₇ cos py + c₈ sin py)

**Case 3: k = −p² < 0.** X″ + p²X = 0 and Y″ − p²Y = 0:

u = (c₉ cos px + c₁₀ sin px)(c₁₁ e^{py} + c₁₂ e^{−py})

Which solution is used depends on the boundary conditions.

#### Part 2: The given problem

**Conditions** (taking the intended form): u(0, y) = 0, u(π, y) = 0, u is bounded as y → ∞, and u(x, 0) = sin x.

**Choice of solution.** The x-boundary conditions are homogeneous at x = 0 and x = π, so X must be oscillatory. This is Case 3:

u = (A cos px + B sin px)(C e^{py} + D e^{−py})

- u(0, y) = 0 gives A = 0.
- u(π, y) = 0 gives B sin pπ = 0, so p = n (integer).
- Bounded as y → ∞ gives C = 0 (e^{ny} would blow up).

u(x, y) = Σ_{n=1}^{∞} bₙ e^{−ny} sin nx

**Condition u(x, 0) = sin x:**

Σ bₙ sin nx = sin x

Comparing coefficients: b₁ = 1, and bₙ = 0 for n ≥ 2.

**Answer:**

**u(x, y) = e^{−y} sin x**

Check: u_xx = −e^{−y} sin x and u_yy = e^{−y} sin x, which sum to 0 ✓. u(0, y) = u(π, y) = 0 ✓. u(x, 0) = sin x ✓. u → 0 as y → ∞ ✓.

## Check yourself

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

1. What are the forms of solution for k=0, k=p^2, and k=-p^2?
   A. All three give trig in x and exponentials in y
   B. Only k = -p^2 gives a solution
   C. Linear (c1x+c2)(c3y+c4); exponentials in x with trig in y; trig in x with exponentials in y
   D. k = 0 gives only u = 0
   Answer: C. X''/X = -Y''/Y = k gives three cases. (Key point: Cases of k)

2. Why is Case 3 used for this problem?
   A. Because y goes to infinity
   B. Because u(x,0) is sin x so Y is trig
   C. Because k = 0 gives no solution
   D. The homogeneous conditions at x=0 and x=pi need oscillatory X, so X is trig
   Answer: D. Trig X can vanish at both x=0 and x=pi. (Key point: Choosing the case)

3. Why is C set to 0?
   A. e^{ny} would blow up as y tends to infinity
   B. To make u(0,y)=0
   C. To satisfy u(pi,y)=0
   D. Because u(x,0) is sin x
   Answer: A. Bounded solutions need the decaying exponential only. (Key point: Boundedness)

4. What is the final solution?
   A. u = e^{y} sin x
   B. u = e^{-y} sin x
   C. u = e^{-y} cos x
   D. u = sin x sinh y
   Answer: B. Only b1 = 1 is nonzero, so u = e^{-y} sin x. (Key point: Final answer)

## Related in Applied Mathematics

- [Laplace equation with u(x,∞) = sin(nπx/l)](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Alaplace-eq-u-x-inf-sin) — Applied Mathematics, chapter 3
- [Laplace equation with u(x,a) = sin(nπx/L)](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Alaplace-eq-u-x-a-sin) — Applied Mathematics, chapter 3, asked 2×
- [Derive 1D heat equation and solve](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aheat-eq-derive-and-solve) — Applied Mathematics, chapter 3, asked 4×
- [Derive 1D wave equation and solve](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Awave-eq-derive-and-solve) — Applied Mathematics, chapter 3, asked 5×
- [Laplace equation in polar form](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Alaplace-eq-polar-form) — 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

- [Derive or solve wave equation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aderive-or-solve-wave-equation) — Mathematics III, chapter 4
- [Solve ODE by Laplace transform (D²−3D+2)y=e^(3t)underth…](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-d2) — Mathematics III, chapter 3
- [Solve ODE by Laplace transform y''+2y'+5y=e^(−t)sint](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-six) — Mathematics III, chapter 3, asked 2×
- [Solve ODE by Laplace transform x''+2x'+5x=e^(−t)sint](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-x) — Mathematics III, chapter 3, asked 17×