Let L be a nonzero real number.

(a) Show that the boundary-value problem y" + λy = 0, y(0) = 0, y(L) = 0 has only the trivial solution y = 0 for the cases λ = C and λ < 0.

(b) For the case λ > 0, find the values of λ for which this problem has a nontrivial solution and give the corresponding solution.

To show: The boundary-value problem y″+λy=0 , y(0)=0 , y(L)=0 has only the trivial solution y=0 for the case λ=0 and the boundary-value problem y″+λy=0 , y(0)=0 , y(L)=0 has only the trivial solution y=0 for the case λ<0 .

Given data:

y″+λy=0 , y(0)=0 , y(L)=0

Formula used:

Write the expression for differential equation.

ay″+by′+cy=0 (1)

Write the expression for auxiliary equation.

ar2+br+c=0 (2)

Write the expression for general solution of ay″+by′+cy=0 with equal roots.

y=c1erx+c2xerx (3)

Write the expression for general solution of ay″+by′+cy=0 with real and distinct.

y=c1er1x+c2er2x (4)

Consider the second order differential equation as follows.

y″+λy=0 (5)

Compare equations (1) and (4).

a=1b=0c=λ

Substitute 1 for a , λ for c and 0 for b in equation (2),

(1)r2+0+λ=0r2+0+λ=0

Substitute 0 for λ ,

r2+0=0r2=0r=0

Substitute 0 for r in equation (3),

y=c1e(0)x+c2xe(0)x

y=c1+c2x (6)

Modify equation (5) as follows.

y(x)=c1+c2x (7)

Substitute 0 for x ,

y(0)=c1+c2(0)=c1

Substitute 0 for y(0) ,

c1=0

Substitute L for x in equation (6),

y(L)=c1+c2L

Substitute 0 for c1 and 0 for y(L) ,

0=0+c2Lc2L=0c2=0

Substitute 0 for c1 and 0 for c2 in equation (5),

y=0+(0)x=0

Thus, the boundary-value problem y″+λy=0 , y(0)=0 , y(L)=0 has only the trivial solution y=0 for the case λ=0 is shown.

Consider the second order differential equation as follows.

y″+λy=0 (8)

Compare equations (1) and (8).

a=1b=0c=λ

Substitute 1 for a , 0 for b and λ for c in equation (2),

(1)r2+(0)r+λ=0r2+λ=0r2=−λr=±−λ

Represent r as r1 and r2 as follows

To find: The values of λ for boundary-value problem, which has a nontrivial solution and give the corresponding solution.