GATE 2025 MA – Question 38

Ordinary Differential Equations · Higher Order Equations and Systems · 2 marks · Multiple choice

Independent solutions of (1+x²)y″−xy′+cos²x y=0 satisfy y₁(0)=3,y₁′(0)=−1,y₂(0)=−5,y₂′(0)=2. Let W=y₁y₂′−y₂y₁′. Find W(1/2).

Show answer and explanation

Correct answer: (B) $\sqrt5/2$

Explanation

W(0)=3×2−(−5)(−1)=1. Abel’s identity gives $W(x)=W(0)\exp[-\int_0^x(-t/(1+t^2))dt]=\sqrt{1+x^2}$. At x=1/2 this is **√5/2, B**.

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