1. The problem statement, all variables and given/known data Using the given differential eq. M'' + bM=0 where b is a real-valued parameter, find all nonzero values for b that adhere to the following boundary conditions: M'(0)=0 M'(Pi)=0 2. Relevant equations When I apply the boundary conditions, after I set up the equation for f(x) using the gen. sol for the characteristic equation M^2+bM=0, do I need to figure out when f(x) has nonzero solutions or when f ' (x) has nonzero solutions? 3. The attempt at a solution ex: when b < 0 taking b= -a where a>0 we get the gen. sol: f(x)=c_1exp(ax) + c_2exp(-ax). applying the bound. conds we find that c_1= c_2 from f '(0)=0 and using f '(Pi)=0 we find that 0=c_1a [exp(aPi)-exp(-aPi)]. If c_1 and c_2 are not equal to 0 we see the exponential part can never be zero, implying c_1 and c_2 are equal to 0. Does this mean there are no nonzero sols for f '(x) which is what we want? and or does this say there are no non-zero sols. for f(x) if that is what we want? I see I need 2 other cases for the value of b. I am just wondering if I am on the right track. Thanks.