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SSC Applied Mathematics QUESTION #1318
Question 1
The eigenvalues for \(X''+\lambda X=0\) with \(X(0)=0\) and \(X(a)=0\) are:
  • \(\lambda_n=\dfrac{n\pi}{a}\)
  • \(\lambda_n=\dfrac{n^2\pi^2}{a^2}\) for \(n=1,2,3,\ldots\)✔️
  • \(\lambda_n=\dfrac{n\pi^2}{a}\)
  • \(\lambda_n=n^2\pi^2\)
Correct Answer Explanation
The general solution is \(X=A\cos(\sqrt{\lambda}\,x)+B\sin(\sqrt{\lambda}\,x)\). From \(X(0)=0\): \(A=0\). From \(X(a)=0\): \(\sin(\sqrt{\lambda}\,a)=0\Rightarrow\sqrt{\lambda_n}=\dfrac{n\pi}{a}\Rightarrow\lambda_n=\dfrac{n^2\pi^2}{a^2}\), with eigenfunctions \(X_n=\sin\!\left(\dfrac{n\pi x}{a}\right)\).