Project ForgeEngineering Mathematics
Calculus
GATE 2017
Hard
Consider the following partial differential equation for u(x,y) with the constant c > 1: ∂u/∂y + c ∂u/∂x = 0. The solution of this equation is
- Au(x,y) = f(x + cy)
- u(x,y) = f(x - cy)
- Cu(x,y) = f(cx + y)
- Du(x,y) = f(cx - y)
Explanation
Try u = f(x-cy). Then u_x = f'(x-cy) and u_y = -c f'(x-cy), so u_y + c u_x = -c f' + c f' = 0, satisfying the PDE identically for any differentiable f.
Source: GATE ME 2017 Session 1 (ME-1) (2017)
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