Question

Consider the BVP

Answer 1

$y''+\lambda y=0$

has the corresponding characteristic equation (CE)

$r^2+\lambda=0$

a. If $\lambda=0$, then the CE has one root, $r=0$, and so the general solution to the ODE is

$y(t)=C_1+C_2t$

Given that $y'(0)=y'\left(\frac\pi6\right)=0$, and

$y'(t)=C_2$

it follows that $C_2=0$, and so

$\boxed{y(t)=C_1}$

b. If $\lambda>0$, then the CE has two complex roots, $r=\pm i\sqrt\lambda$, and the general solution is

$y(t)=C_1\cos(\lambda t)+C_2\sin(\lambda t)$

$\implies y'(t)=-\lambda C_1\sin(\lambda t)+\lambda C_2\cos(\lambda t)$

With the given boundary values, we have

$y'(0)=0\implies\lambda C_2=0\implies C_2=0$

$y'\left(\dfrac\pi6\right)=0\implies-\lambda C_1\sin\left(\dfrac{\lambda\pi}6\right)=0$

$\implies\sin\left(\dfrac{\lambda\pi}6\right)=0$

$\implies\dfrac{\lambda\pi}6=n\pi$

$\implies\lambda=6n$

where $n\in\Bbb Z$.

• If $\lambda$ is a (positive) multiple of 6, we have

$y'\left(\dfrac\pi6\right)=0\implies-6nC_1\sin\left(\dfrac{6n\pi}6\right)=0\implies C_1=0$

and the solution would be

$\boxed{y(t)=0}$

• Otherwise, if $\lambda$ is not a multiple of 6, we have

$y'\left(\dfrac\pi6\right)=0\implies-\lambda C_1\sin\left(\dfrac{\lambda\pi}6\right)=0\implies C_1=0$

so that we still get

$\boxed{y(t)=0}$

c. If $\lambda$, then the CE has two real roots, $r=\pm\sqrt\lambda$, so that the general solution is

$y(t)=C_1e^{\sqrt\lambda\,t}+C_2e^{-\sqrt\lambda\,t}$

$\implies y'(t)=C_1\sqrt\lambda\,e^{\sqrt\lambda\,t}-C_2\sqrt\lambda\,e^{-\sqrt\lambda\,t}$

From the boundary conditions we get

$y'(0)=0\implies C_1\sqrt\lambda-C_2\sqrt\lambda=0\implies C_1=C_2$

$y'\left(\dfrac\pi6\right)=0\implies C_1\sqrt\lambda\,e^{(\pi\sqrt\lambda)/6}-C_2\sqrt\lambda\,e^{-(\pi\sqrt\lambda)/6}=0\implies C_1e^{(\pi\sqrt\lambda)/3}=C_2$

from which it follows that $C_1=C_2=0$, so again the solution is

$\boxed{y(t)=0}$

d. We only get eigenvalues in the case when $\lambda>0$, as in part (b):

$\boxed{\lambda=6n,\,n\in\{1,2,3,\ldots\}}$

for which we get the corresponding eigenfunctions

$\boxed{y(t)=\cos(6nt),\,n\in\{1,2,3,\ldots\}}$