Chapter 9: Q2P (page 481)

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

Short Answer

Expert verified

The first integral of the Euler equation is $x' = \frac{{Cy}^{2}}{\sqrt{1 - C^{2} y^{4}}}$

Step by step solution

Given Information

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

Definition of Euler equation

For the integral, $I (ε) = \int_{x_{1}}^{x_{2}} F (x, y, y') dx$ the Euler equation defined as $\frac{d}{dy} \frac{\partial F}{\partial x'} - \frac{\partial F}{\partial x} = 0$

Find Euler equation of the given function

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

First let’s change the variables as follows

$dx = x' dy y' = \frac{1}{x'}$

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

$\int \frac{\sqrt{1 + y'^{2}}}{y^{2}} dx = \int \frac{\sqrt{1 + y'^{2}}}{y^{2}} x' dy = \int \frac{\sqrt{1 + x'^{‐ 2}}}{y^{2}} x' dy = \int \frac{\sqrt{x'^{2} + 1}}{y^{2}} dy$

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

By the definition of Euler equation $\frac{d}{dy} \frac{\partial F}{\partial x'} - \frac{\partial F}{\partial x'} = 0$

Differentiate $F$ with respect to $x'$ and $x$

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

The Euler become

$\frac{d}{dy} [\frac{1}{y^{2} \sqrt{1 + x'^{2}}}] = 0 \frac{1}{y^{2} \sqrt{1 + x'^{2}}} = C 1 + x'^{2} = \frac{1}{C^{2} y^{4}}$

Therefore, the differential equation is $x' = \pm \frac{{Cy}^{2}}{\sqrt{1 - C^{2} y^{4}}}$

Since minus can be absorbed into the constant.

$x' = \frac{{Cy}^{2}}{\sqrt{1 - C^{2} y^{4}}}$

Therefore, the first integral of the Euler equation is $x' = \frac{{Cy}^{2}}{\sqrt{1 - C^{2} y^{4}}}$

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

Short Answer

Step by step solution

Given Information

Definition of Euler equation

Find 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.2.∫x1x21+y'2y2dx

Short Answer

Step by step solution

Given Information

Definition of Euler equation

Find Euler equation of the given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Dynamics

Linear Momentum

Kinematics Physics

Famous Physicists

Magnetism and Electromagnetic Induction

Engineering Physics

Study anywhere. Anytime. Across all devices.

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