Chapter 9: Q10P (page 478)

Write and solve the Euler equations to make the following integrals stationary. In solving the Euler equations, the integrals in Chapter 5, Section 1, may be useful.
$10 . \int_{x_{1}}^{x_{2}} \frac{x^{2}}{x y' + 1} d x$

Short Answer

Expert verified

Answer

The solution to the Euler equation is $y = D x^{\frac{3}{2}} - I n (x) + B$ .

Step by step solution

Given Information

The given function is $F (x, y, y') = \frac{x^{2}}{x y' + 1}$

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

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

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

Differentiate F with respect to y' and y.

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

The Euler equation becomes:

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

Solve the obtained Euler equation

Now solve the equation $\frac{x^{3}}{{(x y' + 1)}^{2}} = C$ for y'.

Takethe square root on both sides of the equation.

$x y' + 1 = \frac{x \frac{3}{2}}{\sqrt{C}} y' \frac{x \frac{1}{2}}{\sqrt{C}} - \frac{1}{x}$

Integrate with respect to x.

$y = \int \frac{x \frac{1}{2}}{\sqrt{C}} d x - \int \frac{d x}{x} y = D x^{\frac{3}{2}} - I n (x) + B$

Here, we have redefined $D = \frac{2}{3 \sqrt{C}}$ .

So, the solution to the Euler equation is $y = D x^{\frac{3}{2}} - I n (x) + B$ .

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Write and solve the Euler equations to make the following integrals stationary. In solving the Euler equations, the integrals in Chapter 5, Section 1, may be useful.
$10 . \int_{x_{1}}^{x_{2}} \frac{x^{2}}{x y' + 1} d x$

Short Answer

Step by step solution

Given Information

Definition of Euler equation

Find the Euler equation of the given function

Solve the obtained Euler equation

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Write and solve the Euler equations to make the following integrals stationary. In solving the Euler equations, the integrals in Chapter 5, Section 1, may be useful.10.∫x1x2x2xy'+1dx

Short Answer

Step by step solution

Given Information

Definition of Euler equation

Find the Euler equation of the given function

Solve the obtained Euler equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Wave Optics

Quantum Physics

Rotational Dynamics

Astrophysics

Work Energy and Power

Physics of Motion

Study anywhere. Anytime. Across all devices.

Write and solve the Euler equations to make the following integrals stationary. In solving the Euler equations, the integrals in Chapter 5, Section 1, may be useful.
$10 . \int_{x_{1}}^{x_{2}} \frac{x^{2}}{x y' + 1} d x$