Chapter 9: Q4P (page 481)

Change the independent variable to simplify the Euler equation, and then find a first integral of it.
$4 . \int_{x_{1}}^{x_{2}} y \sqrt{y'^{2} + y^{2}} d x$

Short Answer

Expert verified

Answer

The first integral of the Euler equation is $x' = \frac{C}{y \sqrt{y^{2} - C^{2}}}$ .

Step by step solution

Given Information

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

Definition of Euler equation

For the integral $I (ε) = F \int_{x_{1}}^{x_{2}} (x, y, y') d x$ , 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 $\int_{x_{1}}^{x_{2}} y \sqrt{y'^{2} + y^{2}} d x$ .

First, let’s change the variables as follows:

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

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

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

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

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

Differentiate F with respect to x' and x.

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

The Euler becomes as shown below.

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

The differential equation is $x' = \frac{C}{y \sqrt{y^{4} - C^{2}}}$

Therefore, the first integral of the Euler equation is $x' = \frac{C}{y \sqrt{y^{4} - C^{2}}}$

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Change the independent variable to simplify the Euler equation, and then find a first integral of it.
$4 . \int_{x_{1}}^{x_{2}} y \sqrt{y'^{2} + y^{2}} d x$

Short Answer

Step by step solution

Given Information

Definition of Euler equation

Find the Euler equation of the given function

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Change the independent variable to simplify the Euler equation, and then find a first integral of it.4.∫x1x2yy'2+y2dx

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

Translational Dynamics

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Study anywhere. Anytime. Across all devices.

Change the independent variable to simplify the Euler equation, and then find a first integral of it.
$4 . \int_{x_{1}}^{x_{2}} y \sqrt{y'^{2} + y^{2}} d x$