Q36E Verify that F鈥�(x,y)聽聽as defi... [FREE SOLUTION]

91影视

Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 2: Q36E (page 65)

Verify that $F 鈥� (x, y)$ as defined by (9) and (10) satisfies conditions (4).

Short Answer

Expert verified

Verified

Step by step solution

Differentiate with respect to x

The function $F (x, y)$ is given as

$\begin{matrix} F (x, y) = {鈭�}_{x_{o}}^{x} M (t, y) dt + g (y) \\ = N (x, y) - \frac{鈭�}{鈭� y} {鈭�}_{x_{o}}^{x} M (t, y) dt \end{matrix}$

To check if the function satisfies the conditions in equation (4), differentiate with respect to.

$\begin{matrix} \frac{鈭�}{鈭� x} F (x, y) = \frac{鈭�}{鈭� x} ({鈭�}_{x_{o}}^{x} M (t, y) dt + g (y)) \\ = \frac{鈭�}{鈭� x} ({鈭�}_{x_{o}}^{x} M (t, y) dt) + \frac{鈭�}{鈭� x} g (y) \end{matrix}$

Apply the Leibniz theorem then

$\frac{鈭�}{鈭� x} F (x, y) = M (x, y) + \frac{鈭�}{鈭� x} g (y)$

Since g(y) is function of only so

$\frac{鈭�}{鈭� x} F (x, y) = M (x, y)$

This is the condition in equation (4).

Find the result by applying the same procedure with respect to y

$\begin{matrix} \frac{鈭�}{鈭� y} F (x, y) = \frac{鈭�}{鈭� y} ({鈭�}_{x_{o}}^{x} M (t, y) dt + g (y)) \\ = \frac{鈭�}{鈭� y} ({鈭�}_{x_{o}}^{x} M (t, y) dt) + \frac{鈭�}{鈭� y} g (y) \\ \frac{鈭�}{鈭� x} F (x, y) = \frac{鈭�}{鈭� y} ({鈭�}_{x_{o}}^{x} M (t, y) dt) + g' (y) \end{matrix}$

The result $g' (y) = N (x, y) - \frac{鈭�}{鈭� y} ({鈭�}_{x_{o}}^{x} M (t, y) dt)$

$g' (y) = N (x, y) - \frac{鈭�}{鈭� y} ({鈭�}_{x_{o}}^{x} M (t, y) dt)$

Therefore, the result is $\begin{array}{l} \frac{鈭�}{鈭� x} F (x, y) = \frac{鈭�}{鈭� y} ({鈭�}_{x_{o}}^{x} M (t, y) dt) + N (x, y) - \frac{鈭�}{鈭� y} ({鈭�}_{x_{o}}^{x} M (t, y) dt) \\ \frac{鈭�}{鈭� x} F (x, y) = N (x, y) \end{array}$

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Verify that $F 鈥� (x, y)$ as defined by (9) and (10) satisfies conditions (4).

Short Answer

Step by step solution

Differentiate with respect to x

Find the result by applying the same procedure with respect to y

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

Probability and Statistics

Applied Mathematics

Statistics

Calculus

Study anywhere. Anytime. Across all devices.

91影视

Verify that F鈥�(x,y)as defined by (9) and (10) satisfies conditions (4).

Short Answer

Step by step solution

Differentiate with respect to x

Find the result by applying the same procedure with respect to y

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

Probability and Statistics

Applied Mathematics

Statistics

Calculus

Study anywhere. Anytime. Across all devices.

Verify that $F 鈥� (x, y)$ as defined by (9) and (10) satisfies conditions (4).