I'll give you a very similar case
In[1]:= s =
DSolveValue[{v''[x] == -2*a, v[0] == v0, v[x0] == v0}, v, x]
Out[1]= Function[{x}, v0 - a x^2 + a x x0]
In[2]:= bc = {u[x, 0] == u0 + s[x], u[x, y0] == u1 + s[x],
u[0, y] == 0, u[x0, y] == 0};
leqn = Laplacian[u[x, y], {x, y}] == 0;
In[4]:= sol =
FullSimplify[
DSolveValue[{leqn, bc}, u[x, y], {x, y}, Assumptions -> x0 > 0]]
Out[4]= Inactive[Sum][(
4 Csch[(\[Pi] y0 K[1])/x0] Sin[1/2 \[Pi] K[1]]^2 Sin[(\[Pi] x K[1])/
x0] ((2 a x0^2 + \[Pi]^2 (u1 + v0) K[1]^2) Sinh[(\[Pi] y K[1])/
x0] + (2 a x0^2 + \[Pi]^2 (u0 + v0) K[1]^2) Sinh[(\[Pi] (-y +
y0) K[1])/x0]))/(\[Pi]^3 K[1]^3), {K[1], 1, \[Infinity]}]
