# Brachistochrone Differential Equation

• Bennigan88
In summary, the conversation discusses solving for y' in a given equation, introducing a substitution for y and using it to find the equation for y'. The final result is that dx = 2k^2\sin^2 t dt.
Bennigan88
The first part of this problem asks me to solve the following for y' :
$$\left( 1 + {y'}^2 \right)y = k^2$$
So I have:
$$1 + {y'}^2 = \frac{k^2}{y}$$
$${y'}^2 = \frac{k^2}{y} - 1$$
$$y' = \sqrt{{\frac{k^2}{y} - 1}}$$

Then I am asked to show that if I introduce the following:
$$y = k^2 \sin^2t$$

Then the equation for y' found above takes the form:
$$2k^2\sin^2t dt = dx$$

My attempt looks like this:
$$y' = \sqrt{ \frac{k^2}{y} - 1}$$
$$y' = \sqrt{ \frac{k^2}{k^2 \sin^2t}-1}$$
$$y' = \sqrt{ \csc^2t - 1 }$$
$$y' = \sqrt{ \cot^2t }$$
$$\frac{dy}{dx} = \cot t$$

At this point I feel like I have gotten off track or that I am following the wrong line of argumentation. Either that or I'm out of steam and I can't see how keep this going and get what I'm being asked for. Any insight would be greatly appreciated.

Bennigan88 said:
$$\frac{dy}{dx} = \cot t$$
You need to use the substitution for y in dy/dx also. What does that give you?

Thank you! I get:

$$\frac{dy}{dx} = \frac{dy}{dt} \frac{dt}{dx}$$
$$\frac{dy}{dx} = 2k^2 \sin t \cos t \frac{dt}{dx}$$
$$\cot t = 2k^2 \sin t \cos t \frac{dt}{dx}$$
$$1 = 2k^2 \sin^2 t \frac{dt}{dx}$$
$$dx = 2k^2 \sin^2 t dt$$

Your genius rivals that of Gauss himself!

## What is a Brachistochrone Differential Equation?

A Brachistochrone Differential Equation is a mathematical equation that describes the path of shortest time between two points in a gravitational field. It is derived from the principle of least action and is used to solve problems in physics and engineering.

## Who discovered the Brachistochrone Differential Equation?

The Brachistochrone Differential Equation was discovered by the Swiss mathematician and physicist Johann Bernoulli in 1696. He presented it as a challenge to other mathematicians, and it was later solved by his brother, Jakob Bernoulli.

## What are the applications of Brachistochrone Differential Equation?

The Brachistochrone Differential Equation has various applications in physics and engineering, including determining the time-optimal path for a falling object, finding the shape of a cable suspended between two points, and optimizing the path of a spacecraft re-entering the Earth's atmosphere.

## How is the Brachistochrone Differential Equation solved?

The Brachistochrone Differential Equation can be solved using the calculus of variations, a mathematical technique for finding the function that minimizes a given functional. In this case, the functional represents the time taken to traverse the path.

## What are the limitations of the Brachistochrone Differential Equation?

The Brachistochrone Differential Equation assumes a frictionless and uniform gravitational field, which is not always the case in real-world scenarios. It also does not take into account other factors such as air resistance and the shape and mass of the object, which can affect the actual time taken to travel between two points.

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