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GATE 2016 CE (CE1) – Question 40

Engineering Mathematics · ODE: First order and higher order linear equations, Euler-Cauchy equations, initial and boundary value problems · 2 marks · Multiple choice

For $y^{(4)}+3y^{\prime\prime}=108x^2$, which complementary function and particular integral are correct?

  1. $c_1+c_2x+c_3\sin(\sqrt3x)+c_4\cos(\sqrt3x);\quad3x^4-12x^2+c$
  2. $c_2x+c_3\sin(\sqrt3x)+c_4\cos(\sqrt3x);\quad5x^4-12x^2+c$
  3. $c_1+c_3\sin(\sqrt3x)+c_4\cos(\sqrt3x);\quad3x^4-12x^2+c$
  4. $c_1+c_2x+c_3\sin(\sqrt3x)+c_4\cos(\sqrt3x);\quad5x^4-12x^2+c$

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Correct answer: (A) $c_1+c_2x+c_3\sin(\sqrt3x)+c_4\cos(\sqrt3x);\quad3x^4-12x^2+c$

Explanation

The auxiliary equation is m⁴+3m²=m²(m²+3)=0, giving complementary terms c1+c2x+c3sin(sqrt3 x)+c4cos(sqrt3 x). A polynomial particular integral is3x⁴−12x²: its fourth derivative is72 and three times its second derivative is108x²−72. Their sum is108x², so the complete solution is **A**.