Euler Lagrange Equation Question

[HiggsBrozon]

Homework Statement

Consider the function f(y,y',x) = 2yy' + 3x²y where y(x) = 3x⁴ - 2x +1. Compute ∂f/∂x and df/dx. Write both solutions of the variable x only.

Homework Equations

Euler Equation: ∂f/∂y - d/dx * ∂f/∂y' = 0

The Attempt at a Solution

Would I first just find ∂f/∂y and ∂f/∂y' as follows:

∂f/∂y = 2y' + 3x²
∂f/∂y' = 2y

and then insert y(x) and y'(x) into my two equations for ∂f/∂y and ∂f/∂y':

y(x) = 3x⁴ - 2x +1
y'(x) = 12x³ - 2

→ ∂f/∂y = 2(12x³ - 2) + 3x²
→ ∂f/∂y' = 2(3x⁴ - 2x +1)

This is where I begin to get lost. Would I plug these values into my Euler equation to find an expression for df/dx and ∂f/∂x?

I would attempt this but I want to see if I'm headed in the right direction or not first. Any help would be much appreciated.

 

[BruceW]

Would I first just find ∂f/∂y and ∂f/∂y' as follows:

You don't need to find those at all. You don't need to use the Euler-Lagrange equations either. Just calculate the things they ask for in the question. I think they are just testing your knowledge of how to do partial differentiation and the full derivative.

 

[HiggsBrozon]

Thanks for you response. I feel as if I made this problem more complicated than it should.

For ∂f/∂x:

∂f/∂x = 6xy = 6x(3x⁴ - 2x +1)

and df/dx:

df/dx = ∂f/∂y * dy/dx
= d/dy(3x²y + 2yy') * (12x³ - 2)
= (3x² + 2(d/dy(yy'))) * (12x³ - 2)
= (3x² + 2(y * d/dy(y') + y')) * (12x³ - 2)

However I'm a bit confused on how to take the derivative of d/dy(y')
Would I just treat y' as a fraction and cancel out dy's like so: y * d/dy(dy/dx) = d/dx * y = y'

 

[BruceW]

I think your answer for ∂f/∂x is correct. But I don't agree with your answer for df/dx. The line: df/dx = ∂f/∂y * dy/dx I think is not right. This is not the correct expression for the full derivative. p.s. you can realize this by thinking what is being held constant for ∂f/∂y

 

[Nickie]

Your approach is correct so far. To find ∂f/∂x and df/dx, you would plug in the values for ∂f/∂y and ∂f/∂y' that you have calculated into the Euler equation:

∂f/∂y - d/dx * ∂f/∂y' = 0

This will give you an expression for df/dx. Then, you can use the chain rule to express df/dx in terms of x only, using the given function y(x). This will give you the final solution for df/dx in terms of x only. Similarly, you can use the chain rule to express ∂f/∂x in terms of x only, using both y(x) and y'(x). This will give you the final solution for ∂f/∂x in terms of x only. Remember to simplify your expressions as much as possible to get a final solution.