Finding a Constant $c$ so that a given ODE has a solution.

Question

Question. Find the constant $c$ such that the following problem has a solution $$-u''=c ~\text{in}~(a,b)\\ u'(a)=-1,~u'(b)=1$$

My Attempt. Integrating the given equation we have:$$u(x)=-c\frac{x^2}{2}+Ax+B;~~ ~ (A,B ~\text{are arbitrary constants.)}$$

Now from the boundary conditions: $$-ca+A=-1\\-cb+A=1$$

Finally Adding last two equations I get: $c=\frac{2A}{a+b}$

Here $A$ is arbitrary it means $c$ is also arbitrary...Is this right...?? Does every $c$ the given equation has a solution satisfying the boundary condition?If I am wrong Please tell me how to do this type of problem.....Please help..!!

Answer 1

Hint: Instead of adding, subtract the two equations obtained from the boundary condidtions.