Project ForgeEngineering Mathematics
Calculus
GATE 2019
Hard
A differential equation is given as x²(d²y/dx²)−2x(dy/dx)+2y=4. The solution of the differential equation in terms of arbitrary constants C1 and C2 is
- y=C1x²+C2x+2
- By=C1/x²+C2x+2
- Cy=C1x²+C2x+4
- Dy=C1/x²+C2x+4
Explanation
This is an Euler-Cauchy equation. Trying y=x^m in the homogeneous part: m(m−1)−2m+2=0 → m²−3m+2=0 → m=1,2, giving y_h=C1x²+C2x. For the particular solution, try y_p=A (constant): 0−0+2A=4 → A=2. So y=C1x²+C2x+2.
Source: GATE ME 2019 Session 2 (Set 3) (2019)
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