Q20E Verify that when the linear diff... [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: Q20E (page 70)

Verify that when the linear differential equation $[P (x) y - Q (x)] dx + dy = 0$ is multiplied by $渭 (x) = e^{鈭� P (x) dx}$ , the result is exact.

Short Answer

Expert verified

Verified

Step by step solution

General form of special integrating factors

Theorem 3:

If $\frac{(\frac{鈭� M}{鈭� y} - \frac{鈭� N}{鈭� x})}{N}$ is continuous and depends only on x, then $渭 (x) = \exp [鈭� (\frac{(\frac{鈭� M}{鈭� y} - \frac{鈭� N}{鈭� x})}{N}) dx]$

is an integrating factor for the equation.

If $\frac{(\frac{鈭� N}{鈭� x} - \frac{鈭� M}{鈭� y})}{M}$ is continuous and depends only on y, then $渭 (y) = \exp [鈭� (\frac{(\frac{鈭� N}{鈭� x} - \frac{鈭� M}{鈭� y})}{M}) dy]$

is an integrating factor for the equation.

Evaluation method

Given that, $[P (x) y - Q (x)] dx + dy = 0 路路路路路路 (1)$

Multiply by $渭 (x) = e^{鈭� P (x) dx}$ on both sides. We get,

$[P (x) {ye}^{鈭� P (x) dx} - Q (x) e^{鈭� P (x) dx}] dx + e^{鈭� P (x) dx} dy = 0 路路路路路路 (2)$ $[P (x) {ye}^{鈭� P (x) dx} - Q (x) e^{鈭� P (x) dx}] dx + e^{鈭� P (x) dx} dy = 0 路路路路路路 (2)$

Let $M = P (x) {ye}^{鈭� P (x) dx} - Q (x) e^{鈭� P (x) dx}, N = e^{鈭� P (x) dx}$ .

Then,

$\begin{array}{l} \frac{鈭� M}{鈭� y} = P (x) e^{鈭� P (x) dx} \\ \frac{鈭� N}{鈭� x} = P (x) e^{鈭� P (x) dx} \end{array}$

Therefore, .

$\frac{鈭� M}{鈭� y} = \frac{鈭� N}{鈭� x}$

So, the result for this problem is exact.

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91影视

Verify that when the linear differential equation $[P (x) y - Q (x)] dx + dy = 0$ is multiplied by $渭 (x) = e^{鈭� P (x) dx}$ , the result is exact.

Short Answer

Step by step solution

General form of special integrating factors

Evaluation method

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Verify that when the linear differential equation Pxy-Qxdx+dy=0is multiplied by 渭x=e鈭�Pxdx, the result is exact.

Short Answer

Step by step solution

General form of special integrating factors

Evaluation method

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Discrete Mathematics

Applied Mathematics

Logic and Functions

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Verify that when the linear differential equation $[P (x) y - Q (x)] dx + dy = 0$ is multiplied by $渭 (x) = e^{鈭� P (x) dx}$ , the result is exact.