Consider an integral equation (Volterra's integral equation of the third kind) $$(d-cx) u(x) = \int_x^b u(y) dy, \qquad x \in [a,b] \qquad (1) $$ where $u:[a,b] \to \mathbb{R}$ is an unknown function and $a,b,c,$ and $d$ are (non-negative) constants such that $d = cb$ (that is, function $d - cx$ vanishes at $x=b$).
It is straightforward to verify that $$ u(x) = h_1 (d-cx)^{\frac{1-c}{c}}, $$ for any constant $h_1$ is a solution to (1). Is this the unique solution?