I need some guidance on solving a very simple ODE of the form : y'' + y' = 0 , y a function of x. This will then be used to solve an inhomogeneous ODE using a Greens function but that is a later step that cannot be addressed until the below problem is resolved. And the problem is that the boundary conditions seem to suggest a trivial solution so I want to check if the problem has been correctly specified or am I missing something.

My two independent solutions are denoted y1 and y2 such that we can call y = y1 + y2 and the boundary conditions are: when x = 0 , y1 = 0 and when x=1 y2 = 0

The solution to the ODE is found to be : y = A + B exp (-x), with A and B constants to be determined from boundary conditions

So according to me, if I set y1 = A and y2 = B exp(-x)

At x = 0, y1 = 0 implies I think that A has to be zero as there is no dependence on x and this is the only way y1 =0

At x = 1 we are told that y2 = 0 , but that suggests B = 0 which makes the whole solution trivial => A=B =0

Alternatively just as a check if we call y1 = B exp (-x) and y2 = A

then at x=0, y1 = 0 which means B=0 and when x=1 y2= 0 again implying that A = 0 => A=B=0 again

This seems to trivialise the whole problem to y = 0 which is obviously a solution but not very interesting

So either I am doing something seriously wrong such as using the wrong solution to the ODE, missing something really key or the problem is mis-specified with incorrect boundary conditions - I am inclined to believe the first two possibilities and am looking for guidance not fully answers -unless you think it is an ill posed question.

Thank you.