Project ForgeEngineering Mathematics
Calculus
GATE 2023
Hard
Consider the second-order linear ODE x²(d²y/dx²)+x(dy/dx)−y=0, x≥1, with initial conditions y(x=1)=6, dy/dx|(x=1)=2. The value of y at x=2 equals ______.
9(± 0.1 accepted)
Explanation
This is an Euler-Cauchy equation; trying y=xᵐ gives m(m−1)+m−1=0 → m²−1=0 → m=±1. General solution y=C1x+C2/x. From y(1)=C1+C2=6 and y'(1)=C1−C2=2: C1=4, C2=2. So y(x)=4x+2/x, and y(2)=8+1=9.
Source: GATE ME 2023 Official Paper (2023)
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