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GATE 2025 ES – Question 12

Mathematics Foundation · Calculus · 1 mark · Multiple choice

Find the general solution of $d^2y/dx^2+4y=0$.

  1. $y=c_1\cos2x+c_2\sin2x$
  2. $y=c_1\cosh2x+c_2\sinh2x$
  3. $y=c_1e^{2x}+c_2e^{-2x}$
  4. $y=c_1e^{2x}\cos2x+c_2e^{-2x}\sin2x$

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Correct answer: (A) $y=c_1\cos2x+c_2\sin2x$

Explanation

The characteristic equation is $r^2+4=0$,with roots $r=\pm2i$. A real fundamental solution pair is therefore $\cos2x,\sin2x$. Their arbitrary linear combination is **A**. Differentiating twice multiplies either by −4,as required.

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