Q6P Write and solve the Euler equati... [FREE SOLUTION]

Chapter 1: Q6P (page 1)

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.
${鈭�}_{x_{1}}^{x_{2}} (y'^{2} + \sqrt{y}) d x$

Short Answer

Expert verified

After solving the Euler equations to make the integral stationary, the answer is ${(x + k)}^{2} = \frac{16}{9} (\sqrt{y} + c) {(\sqrt{y} - 2 c)}^{2}$ .

Step by step solution

Given Information.

The given integral is $I = {鈭�}_{x_{1}}^{x_{2}} (y'^{2} + \sqrt{y}) d x$ .

Definition/ Concept.

The formulas related to this questions are $鈭� \frac{d x}{\sqrt{x^{2} + a}} = \sinh^{- 1} \frac{x}{a}$ , $鈭� \frac{d y}{\sqrt{y^{2} - a^{2}}} = \cosh^{- 1} \frac{y}{a}$ etc.

Solve the Euler equation to make integral stationary.

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

Then:

$\frac{鈭� F}{鈭� y} = \frac{1}{2 \sqrt{y}} \frac{鈭� F}{鈭� y'} = 2 y'$

By Euler鈥檚 equation $\frac{d}{d x} (\frac{鈭� F}{鈭� y'}) - \frac{鈭� F}{鈭� y} = 0$ , we have:

$\frac{d}{d x} (\frac{鈭� F}{鈭� y'}) - \frac{鈭� F}{鈭� y} = 0 \frac{d}{d x} (2 y') - \frac{1}{2 \sqrt{y}} = 0 2 y'' - \frac{1}{2 \sqrt{y}} = 0$

Let $y' = p . So,$

$y'' = \frac{d p}{d x} y'' = \frac{d p}{d y} . \frac{d y}{d x} y'' = p \frac{d p}{d y}$

So:

$2 p \frac{d p}{d y} - \frac{1}{2 \sqrt{y}} = 0 2 p d p - \frac{d y}{2 \sqrt{y}} = 0$

Now integrating the above equation as:

$p^{2} - \frac{1}{2} \frac{y^{- \frac{1}{2} + 1}}{2 (\frac{1}{2})} = c p^{2} = c + \sqrt{y} p = \sqrt{c + \sqrt{y}} \frac{d y}{d x} = \sqrt{c + \sqrt{y}}$

Now by the variable separable DE, we can write:

$\frac{d y}{d x} = \sqrt{c + \sqrt{y}} 鈭� \frac{d y}{\sqrt{c + \sqrt{y}}} = 鈭� d x + k$

Let $y = t^{2}, d y = 2 t d t$ , we get:

$x + k = 2 鈭� \frac{t}{\sqrt{c + t}} d t x + k = 2 鈭� \frac{t + c - c}{\sqrt{c + t}} d t x + k = 2 [鈭� \sqrt{c +} t d t - 鈭� \frac{c}{\sqrt{c + t}} d t] x + k = 2 脳 \frac{{(c + t)}^{\frac{3}{2}}}{\frac{3}{2}} - 2 c 脳 \frac{{(c + t)}^{- \frac{1}{2} + 1}}{\frac{1}{2}}$

Solving further as:

$x + k = \frac{4}{3} {(c + t)}^{\frac{3}{2}} - 4 c {(c + t)}^{\frac{1}{2}} x + k = \frac{4}{3} {(c + t)}^{\frac{1}{2}} (t - 2 c) {(x + k)}^{2} = \frac{16}{9} (\sqrt{y} + c) {(\sqrt{y} - 2 c)}^{2}$

Therefore, the answer is ${(x + k)}^{2} = \frac{16}{9} (\sqrt{y} + c) {(\sqrt{y} - 2 c)}^{2}$ .

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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.
${鈭�}_{x_{1}}^{x_{2}} (y'^{2} + \sqrt{y}) d x$

Short Answer

Step by step solution

Given Information.

Definition/ Concept.

Solve the Euler equation to make integral stationary.

One App. One Place for Learning.

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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.鈭�x1x2(y'2+y)dx

Short Answer

Step by step solution

Given Information.

Definition/ Concept.

Solve the Euler equation to make integral stationary.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Famous Physicists

Wave Optics

Measurements

Oscillations

Dynamics

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.
${鈭�}_{x_{1}}^{x_{2}} (y'^{2} + \sqrt{y}) d x$