# Calculus of Variations - more confusion

• insynC
In summary, the conversation discusses a question about Calculus of Variation and the attempt at solving the Euler equation for a given integral. The speaker is unsure if they are applying the method correctly and asks for clarification. They receive helpful advice on how to approach the problem and solve for y'.
insynC
I asked a question earlier about Calculus of Variation, but the question I gave didn't really highlight my confusion well. I've come across some other questions that I think reveal my misunderstanding.

## Homework Statement

Solve the Euler equation for the following integral:

(integral from x1->x2) ∫[(y')² + √y]dx

## Homework Equations

Euler equation:

∂F/∂y - d/dx (∂F/∂y') = 0

## The Attempt at a Solution

F = (y')² + √y

So ∂F/∂y = 1/[2√y] and d/dx (∂F/∂y') = 2y'

Thus: y'' = 1/[4√y]

Although it has been some time since I took an ODE course, I think that the equation above is non-trivial to solve. So either I'm mistaken and this is easy to solve or I'm going about the calculus of variations method mistakenly.

With a similar problem, ∫[1+yy']²dx, I was only able to reduce it down to: y''y² + y(y')² = 0, which again I couldn't solve.

I'm not looking for answers, just want to know where I'm applying the method incorrectly, or if in fact I'm missing a far easier way to apply it.

Thanks for any help!

Any thoughts?

Is y'' = 1/[4√y] easily solvable? If not how else can the method be applied?

Cheers

As far as calculus of variations goes, they look fine. Just because you get a difficult ODE doesn't mean the method is wrong.

insynC said:
Any thoughts?

Is y'' = 1/[4√y] easily solvable? If not how else can the method be applied?

Cheers

Try solving for y' first by noting that

$$y''(x)=\frac{d}{dx}y'(x)=\left(\frac{d}{dy}\frac{dy}{dx}\right)y'(x)=\frac{dy'}{dy}y'$$

Last edited:
Thanks gabbagabbahey, that was a big help.

## 1. What is the purpose of Calculus of Variations?

The purpose of Calculus of Variations is to find the function or curve that minimizes or maximizes a given functional. This is useful in many real-world applications, such as optimizing shapes for engineering designs or finding the path that minimizes energy usage.

## 2. What is the difference between Calculus of Variations and traditional Calculus?

Calculus of Variations deals with functionals, which are functions of functions, rather than just functions. It focuses on finding the function that minimizes or maximizes the functional, rather than finding the derivative of a function.

## 3. What is the Euler-Lagrange equation?

The Euler-Lagrange equation is a necessary condition for a function to be a solution to a variational problem. It is derived from the principle of stationary action, which states that the true path of a particle is the one that minimizes the action integral. The equation is given by dF/dy - d/dx(dF/dy') = 0, where F is the functional, y is the function, and y' is the derivative of the function.

## 4. How is Calculus of Variations applied in real life?

Calculus of Variations is applied in many real-life situations, including optimization problems in engineering, economics, and physics. It is also used in the study of optimal control theory, which deals with finding the best way to control a system to achieve a specific goal.

## 5. What are some common challenges when working with Calculus of Variations?

Some common challenges when working with Calculus of Variations include dealing with complex integrals, finding boundary conditions for which the functional is minimized or maximized, and understanding the physical interpretation of the solutions. It also requires a strong understanding of traditional Calculus and mathematical concepts such as Lagrange multipliers and partial derivatives.

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