Hi. There are plenty of methods to solve your problem. Perhaps, the simplest one is by using Finite Difference; you can find many excellent textbooks on this topics. In particular, I suggest Numerical approximation of partial differential equations by by Quarteroni and Valli.
That's being said, with these boundary conditions the analytical solution is 
because: 
, and obviously 
hence: 
.
because: , and obviously hence: . Assuming that indeed you want to solve: 
here is a small code which executes the computation. The idea is to approximate: 
, where 
, 
= nodes of the compuational domain, and 
= discretization step.
here is a small code which executes the computation. The idea is to approximate: , where , = nodes of the compuational domain, and = discretization step.Hence you can discretize your PDE, and this boils up into a linear system:
together with imposed Dirichlet boundary conditions: 
% Parameters of the PDE
% ---------------------
c1 = 1;
H = 2;
P = 1;
b = @(y) 1+y; % Source term (it has to be different from 0)
m0 = 0; % Value of m(0)
mH = 0; % Value of m(H)
% Parameters of the solver
% ------------------------
N = 100; % Number of nodes
% Finite Difference solver
% ------------------------
A = sparse(N, N); % Matrix of the linear system
z = zeros(N, 1); % Known vector
h = H/(N-1); % = Discretization step
y = linspace(0, H, N); % = Coordinates of the nodes
bn = feval(b, y); % = Source term b(y) computed in each node
% Assembly the laplacian operator
one = ones(N, 1);
A = spdiags([one -2*one one]./h^2, -1:1, N, N);
% Assembly the term -c1*m
A = A + spdiags(-c1*one, 0, N, N);
% Assembly the source term
z = c1*P./bn(:);
% Impose Dirichlet boundary conditions by replacing the first and last
% equations of the linear system
A(1,:) = 0 ; A(1,1) = 1 ; z(1) = m0; % impose m(1) = 0
A(N,:) = 0 ; A(N,N) = 1 ; z(N) = mH; % impose m(N) = 0
% Solve the linear system and plot the result
m = A \ z;
figure
plot(y, m, '.-');
xlabel('y') ; ylabel('m(y)');
You can modify the code so as to check it with more trivial problems, for instance: 
with 
and 
, which will provide the (expected) solution 
.
with and , which will provide the (expected) solution . Enjoy.
