Q. Obtain the partial differential equation by eliminating the arbitrary constant from the relation u=ax+(1-a)y+b
There are two arbitrary constants here, a and b.
Differentiating u wrt x,
ꝺu/ꝺx = a
Differentiating u wrt y,
ꝺu/ꝺy = (1-a)
ꝺu/ꝺy + a = 1
Substituting for a = ꝺu/ꝺx,
ꝺu/ꝺx + ꝺu/ꝺy = 1