Discussion
Consider the functional
\(J[y] = \displaystyle\int_{a}^{b} F(x, y, y')\, dx\)
where \(F(x, y, y') = \dfrac{1 + y^{2}}{y'^{2}}\)
for admissible function \(y(x)\). Which of the following are extremals for J ?
Answer: C
The extremals are \(y(x) = \sinh(Ax + B)\).
F does not contain \(x\), so the Euler-Lagrange equation has the first integral \(F - y'F_{y'} = c\):
\[ \begin{aligned} F_{y'} &= -\frac{2(1 + y^{2})}{y'^{3}} \\ F - y'F_{y'} &= \frac{3(1 + y^{2})}{y'^{2}} = c \end{aligned} \]
So \(y'^{2} = A^{2}(1 + y^{2})\), that is \(dy/\sqrt{1 + y^{2}} = A\,dx\), giving \(\sinh^{-1} y = Ax + B\) and \(y = \sinh(Ax + B)\).
The trigonometric families solve \(y'' = -y\), and \(A\sinh x + B\cosh x\) solves \(y'' = y\); neither satisfies \(y'^{2} = A^{2}(1 + y^{2})\) in general.
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