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CSS Applied Mathematics QUESTION #1317
Question 1
Assuming \(u(x,y)=X(x)Y(y)\) in \(u_{xx}+u_{yy}=0\), the separated ODEs are:
  • \(X''-\lambda X=0\) and \(Y''+\lambda Y=0\)
  • \(X''+\lambda X=0\) and \(Y''-\lambda Y=0\)✔️
  • \(X''+\lambda X=0\) and \(Y''+\lambda Y=0\)
  • \(X''-\lambda X=0\) and \(Y''-\lambda Y=0\)
Correct Answer Explanation
Substituting: \(X''Y+XY''=0\Rightarrow\dfrac{X''}{X}=-\dfrac{Y''}{Y}=\lambda\). For the \(x\)-direction with \(u(0,y)=u(a,y)=0\), we need oscillatory solutions: \(X''+\lambda X=0\) with \(\lambda>0\). Then \(Y''-\lambda Y=0\) gives exponential/hyperbolic solutions.