Given a functional $J[y]=\int_a^b (y^2+(y')^2-2ye^x) \ dx$. Investigate whether the interval $[a,b]$ contains conjugate points of $a$.
Attempt: Let $F(x,y,y')=y^2+(y')^2-2ye^x$, we have $F_{y}=2y-2e^x$ and $F_{y'}=2y'$, so that $F_{yy}=2, F_{y'y'}=2$, and $F_{yy'}=0$. Now, we have $P=\frac{1}{2}F_{y'y'} = 1$ and $Q=\frac{1}{2}\left(F_{yy} - \frac{d}{dx}F_{yy'} \right) = 1$. Thus, we have the Riccati equation \begin{equation*} -\frac{d}{dx}(Ph')+Qh=0 \iff h''-h=0. \end{equation*} Hence, the general solution of this Riccati equation is $h(x)=Ae^{-x}+Be^x$, where $A$ and $B$ are arbitrary constant. Let $h_1(x)=e^{-x}$ and $h_2(x)=e^x$. The Wronskian of $h_1$ and $h_2$ is $2 \ne 0$, and so, $h_1$ and $h_2$ are linearly independent on $[a,b]$. Clearly, only the trivial solution can satisfy the conditions $h(a)=0$ and $h(K)=0$, for any $K \in \Bbb R - \{a\}$. Hence, there are no points in the $[a,b]$ which are conjugate to $a$.
The other way to investigate whether the interval $[a,b]$ contains conjugate points of $a$ is by considering that for any conjugate points $K$ to $a$ must satisfy \begin{equation*} h_2(K)h_1(0) = h_1(K)h_2(0) \end{equation*} i.e. \begin{equation*} e^{2K} = e^{2a}, \end{equation*} which gives $K=a$. By definition of a conjugate point, $K$ must be not equal to $a$. Therefore, there are no points in the interval $[a,b]$ which are conjugate to $a$.
Am I correct? Any comments and corrections are appreciated. Thanks in advanced.
The references I used:
- Gelfand, I. M., Fomin, S.V. - Calculus of Variations, page 106-114.
- Bruce van Brunt - The Calculus of Variations, page 235-236 and 245-249.