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GATE 2017 ME (ME1) – Question 14

Engineering Mathematics · Differential Equations: First order and higher order linear equations, Euler-Cauchy equation · 1 mark · Multiple choice

The equation $d^2y/dx^2+16y=0$ has boundary conditions $y\prime(0)=1$ and $y\prime(\pi/2)=-1$. It has

  1. No solution.
  2. Exactly two solutions.
  3. Exactly one solution.
  4. Infinitely many solutions.

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Correct answer: (A) No solution.

Explanation

The general solution is $y=A\cos4x+B\sin4x$, with derivative $y\prime=-4A\sin4x+4B\cos4x$. Both at $x=0$ and at $x=\pi/2$, this derivative equals $4B$. It cannot equal both 1 and −1. The boundary conditions are inconsistent, so there is no solution, A.

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