Chapter 16, Problem 7PS

(a)

To determine

To prove: That the differential equation $y^{″}+ay=0$ with the boundary conditions $y(0)=0\text{and}y(L)=0$ has only the trivialsolution $y=0$ for $a=0$.

(b)

To determine

To prove: That the differential equation $y^{″}+ay=0$ with the boundary conditions $y(0)=0\text{and}y(L)=0$ has only the trivial solution $y=0$ for $a<0$.

(c)

To determine

To prove: That the differential equation $y^{″}+ay=0$ with the boundary conditions $y(0)=0\text{and}y(L)=0$ has non trivial solutions for $a>0$. Also, find the solutions.