Question

Follow the standard algorithm: • Write down the Euler-Lagrange equation. • Obtain the solution $y(x)$ of the Euler-Lagrange equation with the given boundary conditions. • Introduce a deviation $\eta(x)$ from the obtained solution $y(x)$ and consider the diffe- rence $J[y(x) + \eta(x)] - J[y(x)]$ in order to see if the function $y(x)$ is a minimum, a maximum or not an extremum of the functional. For example, if $J[y(x) + \eta(x)] -$ $J[y(x)] > 0$ for all nonzero $\eta(x)$, then $y(x)$ is the absolute minimum of the func- tional. Hint: solutions of differential equations of the form $Ax^2y'' + Bxy' + Cy = 0$ may be sought in the form of power-law functions $y(x) \propto x^\alpha$ (you have to identify possible powers $\alpha$ for a particular equation). 4. Find all the extrema (local minima and maxima) of the functional $$ J[y] = \int_1^2 (xy + y')^2 dx; \quad y(1) = 1, y(2) = \frac{1}{2} $$ Hint: once you've found the solution of the Euler-Lagrange equation with the boun- dary conditions, remember to check, like in the previous problem, if this solution is a minimum, a maximum or not an extremum.

          Follow the standard algorithm:
• Write down the Euler-Lagrange equation.
• Obtain the solution $y(x)$ of the Euler-Lagrange equation with the given boundary
conditions.
• Introduce a deviation $\eta(x)$ from the obtained solution $y(x)$ and consider the diffe-
rence $J[y(x) + \eta(x)] - J[y(x)]$ in order to see if the function $y(x)$ is a minimum, a
maximum or not an extremum of the functional. For example, if $J[y(x) + \eta(x)] -$
$J[y(x)] > 0$ for all nonzero $\eta(x)$, then $y(x)$ is the absolute minimum of the func-
tional.
Hint: solutions of differential equations of the form $Ax^2y'' + Bxy' + Cy = 0$ may be
sought in the form of power-law functions $y(x) \propto x^\alpha$ (you have to identify possible
powers $\alpha$ for a particular equation).
4. Find all the extrema (local minima and maxima) of the functional
$$
J[y] = \int_1^2 (xy + y')^2 dx; \quad y(1) = 1, y(2) = \frac{1}{2}
$$
Hint: once you've found the solution of the Euler-Lagrange equation with the boun-
dary conditions, remember to check, like in the previous problem, if this solution is a
minimum, a maximum or not an extremum.

Follow the standard algorithm:
• Write down the Euler-Lagrange equation.
• Obtain the solution y(x) of the Euler-Lagrange equation with the given boundary
conditions.
• Introduce a deviation η(x) from the obtained solution y(x) and consider the diffe-
rence J[y(x) + η(x)] - J[y(x)] in order to see if the function y(x) is a minimum, a
maximum or not an extremum of the functional. For example, if J[y(x) + η(x)] -
J[y(x)] > 0 for all nonzero η(x), then y(x) is the absolute minimum of the func-
tional.
Hint: solutions of differential equations of the form Ax^2y” + Bxy' + Cy = 0 may be
sought in the form of power-law functions y(x) ∝ x^α (you have to identify possible
powers α for a particular equation).
4. Find all the extrema (local minima and maxima) of the functional

J[y] = ∫1^2 (xy + y')^2 dx; y(1) = 1, y(2) = (1)/(2)

Hint: once you've found the solution of the Euler-Lagrange equation with the boun-
dary conditions, remember to check, like in the previous problem, if this solution is a
minimum, a maximum or not an extremum.

Added by Angela S.

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