Project ForgeEngineering Mathematics
Calculus
GATE 2013
Hard
The solution to the differential equation d²u/dx² − k(du/dx) = 0, where k is a constant, subjected to the boundary conditions u(0)=0 and u(L)=U, is
- Au = Ux/L
- u = U(1−e^(kx))/(1−e^(kL))
- Cu = U(1−e^(−kx))/(1−e^(−kL))
- Du = U(1+e^(kx))/(1+e^(kL))
Explanation
The characteristic equation r²−kr=0 gives r=0 or r=k, so u(x)=A+Be^(kx). u(0)=0 gives A=−B. u(L)=U gives B(e^(kL)−1)=U, so B=U/(e^(kL)−1). Thus u(x)=U(e^(kx)−1)/(e^(kL)−1), which (multiplying numerator and denominator by −1) equals U(1−e^(kx))/(1−e^(kL)).
Source: GATE ME 2013 (Booklet Code A) (2013)
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