Chapter 9: Q3P (page 481)

Change the independent variable to simplify the Euler equation, and then find a first integral of it.
$3 . \int_{x_{1}}^{x_{2}} \frac{x^{t 2}}{\sqrt{x^{t 2} + x^{2}}} d y$

Short Answer

Expert verified

Answer

The equation $x^{4} y'^{2} = C^{2} {(1 + x^{2} y'^{2})}^{3}$ can be left in this form, since the solution is not trivial and there is no need to solve it.

Step by step solution

Given Information

The given integral is $\int_{x_{1}}^{x_{2}} \frac{x'^{2}}{\sqrt{x'^{2} + x^{2}}} d y$ .

Definition of Euler equation

For the integral $l (ε) = \int_{x_{1}}^{x_{2}} (x, y, y) d x$ , the Euler equation is mathematically presented as $\frac{d}{d x} \frac{d F}{d y'} - \frac{d F}{d y} = 0$ .

Find the Euler equation of the given function

The given integral is $x^{4} y'^{2} = C^{2} {(1 + x^{2} y'^{2})}^{3}$ .

First, let’s change the variables as follows:

$d x = x' d y y' = \frac{1}{x'}$

By using the two equations above, we can rewrite the integral.

$\int \frac{x'^{2}}{\sqrt{x'^{2} + x^{2}}} d y = \int \frac{x'^{2}}{\sqrt{x'^{2} + x^{2}}} \frac{d x}{x'} = \int \frac{y'^{- 2}}{\sqrt{y'^{- 2} + x^{2}}} y' d x = \int \frac{1}{y' \sqrt{y'^{- 2} + x^{2}}} d x = \int \frac{1}{\sqrt{1 + y'^{2} x^{2}}} d x$

Let $F (x, y, y') = \frac{1}{\sqrt{1 + y'^{2} x^{2}}}$

By the definition of the Euler equation, $\frac{d}{d x} \frac{\partial F}{\partial F} - \frac{\partial F}{\partial y} = 0$ .

Differentiate F with respect to y' and y.

$\frac{\partial F}{\partial y'} = \frac{y' x^{2}}{{(1 + x^{2} y'^{2})}^{\frac{3}{2}}} \frac{\partial F}{\partial y} = 0$

The Euler equations becomes:

$\frac{d}{d x} [\frac{y' x^{2}}{{(1 + x^{2} y'^{2})}^{\frac{3}{2}}}] = 0 \frac{y' x^{2}}{{(1 + x^{2} y'^{2})}^{\frac{3}{2}}} = C x^{4} y'^{3} {(1 + x^{2} y'^{2})}^{3}$

We shall leave the equation $x^{4} y'^{3} {(1 + x^{2} y'^{2})}^{3}$ in this form, since the solution is not trivial and there is no need to solve it.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Change the independent variable to simplify the Euler equation, and then find a first integral of it.
$3 . \int_{x_{1}}^{x_{2}} \frac{x^{t 2}}{\sqrt{x^{t 2} + x^{2}}} d y$

Short Answer

Step by step solution

Given Information

Definition of Euler equation

Find the Euler equation of the given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Particle Model of Matter

Waves Physics

Circular Motion and Gravitation

Scientific Method Physics

Kinematics Physics

Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.

Change the independent variable to simplify the Euler equation, and then find a first integral of it.3.∫x1x2xt2xt2+x2dy

Short Answer

Step by step solution

Given Information

Definition of Euler equation

Find the Euler equation of the given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Particle Model of Matter

Waves Physics

Circular Motion and Gravitation

Scientific Method Physics

Kinematics Physics

Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.

Change the independent variable to simplify the Euler equation, and then find a first integral of it.
$3 . \int_{x_{1}}^{x_{2}} \frac{x^{t 2}}{\sqrt{x^{t 2} + x^{2}}} d y$