Project ForgeEngineering Mathematics
Differential Equations
GATE 2013
Medium
The solution of the differential equation d²y/dx² − dy/dx + 0.25y=0, given y=0 at x=0 and dy/dx=1 at x=0, is
- Axe^(0.5x) − xe^(−0.5x)
- B0.5xeˣ − 0.5xe⁻ˣ
- xe^(0.5x)
- D−xe^(0.5x)
Explanation
Characteristic equation: r²−r+0.25=0 ⟹ (r−0.5)²=0, a repeated root r=0.5. General solution y=(C1+C2x)e^(0.5x). y(0)=0 ⟹ C1=0. y'=C2e^(0.5x)+0.5(C1+C2x)e^(0.5x); y'(0)=C2=1. So y=xe^(0.5x).
Source: GATE CH 2013 (2013)
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