Find a general solution to Euler's equation for an integral

Question

The question asks me to find the general solution to Euler's equation for:

$$\int^b_ay'^2-8xy+x \, dx.$$

How would one find such a solution? My textbook does not do a very good job of explaining how to solve such a question.

Answer 1

Let $$L(x, y, y') = y'^2 - 8xy + x$$ Then the Euler equation tells us that $$J[y] = \int_a^b L(x,y,y') \, dx$$ has an extremum when $$\frac{d}{dx}\frac{\partial L}{\partial y'} - \frac{\partial L}{\partial y} = 0$$ We can easily calculate these derivatives given $L$: $$\frac{\partial L}{\partial y'} = 2y' \implies \frac{d}{dx}\frac{\partial L}{\partial y} = 2y''$$ and $$\frac{\partial L}{\partial y} = -8x$$ Hence, $J[y]$ has an extremum when:

$$2y'' + 8x = 0$$

Integrating twice, we get:

$$y = -\frac23 x^3 + c_1x + c_2$$

Now, if given boundary conditions (like $y(a) = y_0, y(b) = y_1$), you could determine the values for the constants $c_1, c_2$.

The above expression for $y$ is what they mean when they say to find a solution to the euler equation.

It's poor (but common) language to say this, when they really mean find the function $y(x)$ that extremizes the given integral by solving the corresponding euler equation