Project ForgeEngineering Mathematics
Calculus
GATE 2012
Medium
Consider the differential equation x²(d²y/dx²) + x(dy/dx) − 4y = 0 with the boundary conditions of y(0)=0 and y(1)=1. The complete solution of the differential equation is
- x²
- Bsin(πx/2)
- Ceˣ sin(πx/2)
- De⁻ˣ sin(πx/2)
Explanation
This is a Cauchy-Euler (equidimensional) equation. Substituting y=xᵐ: m(m−1)+m−4=0 → m²−4=0 → m=±2. General solution y=C1x²+C2x⁻². Since y must stay finite (bounded) as x→0, C2=0, leaving y=C1x². Applying y(1)=1 gives C1=1, so y=x² — which also automatically satisfies y(0)=0.
Source: GATE ME 2012 (Booklet Code A) (2012)
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