Project ForgeEngineering Mathematics
Differential Equations
GATE 2012
Medium
If a and b are arbitrary constants, then the solution to the ordinary differential equation d²y/dx²−4y=0 is
- Ay=ax+b
- By=ae⁻ˣ
- Cy=a sin(2x)+b cos(2x)
- y=a cosh(2x)+b sinh(2x)
Explanation
Characteristic equation r²−4=0 gives real roots r=±2, so the general solution is a linear combination of e^(2x) and e^(−2x) — equivalently expressed as y=a·cosh(2x)+b·sinh(2x), since cosh and sinh are themselves linear combinations of e^(2x) and e^(−2x).
Source: GATE CH 2012 (2012)
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