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GATE 2017 EC (EC2) – Question 12

Engineering Mathematics · Differential Equations · 1 mark · Multiple choice

The general solution of the differential equation $d^2y/dx^2+2dy/dx-5y=0$, in terms of arbitrary constants $K_1$ and $K_2$, is

  1. $K_1e^{(-1+\sqrt6)x}+K_2e^{(-1-\sqrt6)x}$
  2. $K_1e^{(-1+\sqrt8)x}+K_2e^{(-1-\sqrt8)x}$
  3. $K_1e^{(-2+\sqrt6)x}+K_2e^{(-2-\sqrt6)x}$
  4. $K_1e^{(-2+\sqrt8)x}+K_2e^{(-2-\sqrt8)x}$

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Correct answer: (A) $K_1e^{(-1+\sqrt6)x}+K_2e^{(-1-\sqrt6)x}$

Explanation

Substitute $y=e^{rx}$ to obtain the characteristic equation $r^2+2r-5=0$.

The quadratic formula gives $r=(-2\pm\sqrt{24})/2=-1\pm\sqrt6$. These are distinct roots, so the two exponential solutions are independent. Their arbitrary linear combination is the expression in **A**.

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