Q5E In Problems , identify the equat... [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: Q5E (page 69)

In Problems , identify the equation as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone.
$(x^{2} sinx + 4 y) dx + xdy = 0$

Short Answer

Expert verified

The given equation is linear and has an integrating factor that is a function of x alone.

Step by step solution

General form of separable, linear, exact or integrating factors

Separable equation: If the right-hand side of the equation $\frac{dy}{dx} = f (x, y)$ can be expressed as a function g(x) that depends only on x times a function p(y) that depends only on y, then the differential equation is called separable.

Linear equation:Standard form of linear equation is $\frac{dy}{dx} + P (x) y = Q (x)$ .

Exact differential form:The differential form $M (x, y) dx + N (x, y) dy$ is said to be exact in a rectangle R if there is a function such that

$\frac{鈭� F}{鈭� x} (x, y) = M (x, y) 鈥� 鈥� 鈥� 鈥� 鈥� and 鈥� 鈥� 鈥� 鈥� \frac{鈭� F}{鈭� y} (x, y) = N (x, y)$

Special integrating factors: If $\frac{(\frac{鈭� M}{鈭� y} - \frac{鈭� N}{鈭� x})}{N}$ is continuous and depends only on x. If $\frac{(\frac{鈭� N}{鈭� x} - \frac{鈭� M}{鈭� y})}{M}$ is continuous and depends only on y.

Evaluate the given equation

Given: $(x^{2} \sin x + 4 y) dx + x dy = 0$

Evaluate:

$\begin{array}{l} (x^{2} \sin x + 4 y) dx + x dy = 0 \\ \frac{dy}{dx} = - \frac{x^{2} \sin x + 4 y}{x} \\ dy = - x \sin x - \frac{4 y}{x} \\ \frac{dy}{dx} + \frac{4 y}{x} = - x \sin x \end{array}$

Compare the equation with general form of separable and linear equation.

So, the given equation is not separable but it is linear.

Testing for exactness

Given: $(x^{2} \sin x + 4 y) dx + x dy = 0$

Let: $M = x^{2} \sin x + 4 y, N = x$

Then:

$\begin{matrix} \frac{鈭� M}{鈭� y} = 4 \\ \frac{鈭� N}{鈭� x} = 1 \end{matrix}$

So $\frac{鈭� M}{鈭� y} 鈮� \frac{鈭� N}{鈭� x}$ .

Therefore, the given equation is not exact.

Computing integrating factor

If $\frac{(\frac{鈭� M}{鈭� y} - \frac{鈭� N}{鈭� x})}{N}$ then the given function is x alone.

If $\frac{(\frac{鈭� N}{鈭� x} - \frac{鈭� M}{鈭� y})}{M}$ then the given function is y alone.

Then, substitute the values to prove it.

$\begin{matrix} \frac{(\frac{鈭� M}{鈭� y} - \frac{鈭� N}{鈭� x})}{N} = \frac{4 - 1}{x} \\ = \frac{3}{x} \end{matrix}$

So, we obtain an integrating factor that is a function of x alone. And the given equation is linear.

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91影视

In Problems , identify the equation as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone.
$(x^{2} sinx + 4 y) dx + xdy = 0$

Short Answer

Step by step solution

General form of separable, linear, exact or integrating factors

Evaluate the given equation

Testing for exactness

Computing integrating factor

One App. One Place for Learning.

Most popular questions from this chapter

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91影视

In Problems , identify the equation as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone.x2sinx+4ydx+xdy=0

Short Answer

Step by step solution

General form of separable, linear, exact or integrating factors

Evaluate the given equation

Testing for exactness

Computing integrating factor

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Decision Maths

Logic and Functions

Probability and Statistics

Statistics

Study anywhere. Anytime. Across all devices.

In Problems , identify the equation as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone.
$(x^{2} sinx + 4 y) dx + xdy = 0$