Finding a function y(x) given three parameters

[DeltaT37]

Homework Statement

This problem comes from a practice test that I am reviewing before my final. My main confusion comes from the mathematical implication of the integral being an extremum. The first two parameters are y(x=0) = 0 and y(x = π/2) = 1. The third says the integral from 0 to 1 of ∫[ (dy/dx)² - y² ]dx is an extremum.

Homework Equations

N/A

The Attempt at a Solution

Clearly, the first two parameters are easily solved by y = sin(x). However, this third bit of information is very confusing to me. I first considered, in 1 dimension the first derivative of a function at an extremum is zero, and thus perhaps plugging in zero for dy/dx. That didn't seem right so I attempted to solve the integral, but couldn't figure out the first part to solve. Thirdly, I tried plugging in sin(x) and see what could be achieved, with a final result of something like .5-.27 = extremum (integral of sin²(x) from zero to one is the .27.) If possible, the guidance I am looking for is how to think about mathematically the fact that the the integrand, involving dy/dx is related to the extremum.

Thanks. Sorry if the formatting is off, first post here.

 

[Ray Vickson]

Homework Statement

View attachment 197657
This problem comes from a practice test that I am reviewing before my final. My main confusion comes from the mathematical implication of the integral being an extremum. The first two parameters are y(x=0) = 0 and y(x = π/2) = 1. The third says the integral from 0 to 1 of ∫[ (dy/dx)² - y² ]dx is an extremum.

Homework Equations

N/A

The Attempt at a Solution

Clearly, the first two parameters are easily solved by y = sin(x). However, this third bit of information is very confusing to me. I first considered, in 1 dimension the first derivative of a function at an extremum is zero, and thus perhaps plugging in zero for dy/dx. That didn't seem right so I attempted to solve the integral, but couldn't figure out the first part to solve. Thirdly, I tried plugging in sin(x) and see what could be achieved, with a final result of something like .5-.27 = extremum (integral of sin²(x) from zero to one is the .27.) If possible, the guidance I am looking for is how to think about mathematically the fact that the the integrand, involving dy/dx is related to the extremum.

Thanks. Sorry if the formatting is off, first post here.

You need to use the Euler-Lagrange equation, to see what the candidate forms can be for $y(x)$ that optimize the integral.

 

[DeltaT37]

You need to use the Euler-Lagrange equation, to see what the candidate forms can be for $y(x)$ that optimize the integral.

Okay thank you very much! This was the advice I was looking for. For future notice, should I assume that when I need to minimize an integral that has dy/dx in it, I should be looking out to use Euler-Lagrange equation? (Apologies again if its a silly question)

 

[Ray Vickson]

Okay thank you very much! This was the advice I was looking for. For future notice, should I assume that when I need to minimize an integral that has dy/dx in it, I should be looking out to use Euler-Lagrange equation? (Apologies again if its a silly question)

If it has both y and dy/dx in it, then yes: the problem of optimizing $\int_a^b F(y(x), y'(x)) \, dx$ belongs to the field of Calculus of Variations.