Home › MCQs › CSS Applied Mathematics › Question #4158
Back to Questions
CSS Applied Mathematics QUESTION #4158
Question 1
For the Laplace equation $u_{xx} + u_{yy} = 0$ on $0 < x < a$, $0 < y < b$ with $u(0,y) = u(a,y) = u(x,0) = 0$ and $u(x,b) = f(x)$, what is the general form of solution using separation of variables?
  • $u(x,y) = \sum_{n=1}^{\infty} B_n \sin(\frac{n\pi x}{a})\sinh(\frac{n\pi y}{a})$✔️
  • $u(x,y) = \sum_{n=1}^{\infty} B_n \cos(\frac{n\pi x}{a})\cosh(\frac{n\pi y}{a})$
  • $u(x,y) = \sum_{n=1}^{\infty} B_n \sin(\frac{n\pi x}{a})\cos(\frac{n\pi y}{b})$
  • $u(x,y) = \sum_{n=1}^{\infty} B_n e^{\frac{n\pi x}{a}}\sin(\frac{n\pi y}{b})$
Correct Answer Explanation
Boundary conditions $u(0,y) = u(a,y) = 0$ suggest $X(x) = \sin(\frac{n\pi x}{a})$. Condition $u(x,0) = 0$ gives $Y(0) = 0$, so $Y(y) = \sinh(\frac{n\pi y}{a})$.