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I have a problem with this first order DE: let $-\infty<a\leq x\leq b<+\infty$ and

$$u'(x)+(\lambda+q(x))u(x)=0,\tag{1}$$ where $u$ is a continuous and real valued, while $\lambda$ is a parameter not depending on $x$.

A strange non trivial boundary condition is given, namely $$\alpha u(a)+\alpha'u'(a)+\beta u(b)+\beta'u'(b)=0.$$

Then I have to show that this problem admits at most three eigenvalues.

What I have tried: basically to convert this problem into a Sturm Liouville problem, however I couldn't conclude anything.

Can anybody help me?

How to go through this kind of problems? thanks in advance.


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I don't know what is meant by an eigenvalue of a problem. – Gerry Myerson Oct 29 '12 at 11:57
Is it $u'$ or $u''$ in the equation? – Julián Aguirre Oct 29 '12 at 13:54
@JuliánAguirre it is $u'$, it is a first order DE – guido giuliani Oct 29 '12 at 16:46
@GerryMyerson I'm following the language introduced in the Birkhoff Rota Ordinary differential equation book and you can define the operator $L[u]=u'+q(x)u$ so for $\lambda$ to be an eigenvalue it means that $L[u]=\lambda u$ has a non trivial solution $u$. – guido giuliani Oct 29 '12 at 16:48
Ah, it's an eigenvalue of an operator, not an eigenvalue of a problem. Got it. Thanks. – Gerry Myerson Oct 29 '12 at 23:30
up vote 1 down vote accepted

The equation is a linear first order differential equation, whose solution is $$ u(x)=C\,e^{-\lambda x-\int_a^xq(t)dt}. $$ We may take $C=1$. Then $$\begin{align*} u(a)&=e^{-\lambda a},\\ u'(a)&=-(\lambda+q(a))e^{-\lambda a},\\ u(b)&=e^{-\lambda b-\int_a^bq(t)dt},\\ u'(b)&=-(\lambda+q(b))e^{-\lambda b-\int_a^bq(t)dt}. \end{align*}$$ Let $k=e^{-\int_a^bq(t)dt}$. The boundary condition is then $$ \alpha\,e^{-\lambda a}-\alpha'(\lambda+q(a))\,e^{-\lambda a}+\beta\,k\,^{-\lambda b}-\beta'\,k\,(\lambda+q(b))e^{-\lambda b}=0, $$ which can be written as the following equation in the unknown $\lambda$: $$ -\alpha'\lambda+k(\beta-\beta'q(b))e^{-\lambda(b-a)}-k\,\beta'\lambda\,e^{-\lambda(b-a)}=\alpha'q(a)-\alpha. $$ You have to study the number of solutions of this equation according to the possible values of the parameters.

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