Chapter 2: Q3E (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.
$(2 x + y) dx + (x - 2 y) dy = 0$

Short Answer

Expert verified

The given equation is exact.

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

$\frac{\partial F}{\partial x} (x, y) = M (x, y) and \frac{\partial F}{\partial y} (x, y) = N (x, y)$

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

Evaluate the given equation

Given, $(2 x + y) dx + (x - 2 y) dy = 0$ .

Evaluate it.

$\begin{matrix} (2 x + y) dx + (x - 2 y) dy = 0 \\ \frac{dy}{dx} = \frac{2 x + y}{x - 2 y} \end{matrix}$

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

So, the given equation is neither separable nor linear.

Testing for exactness

Given, $(2 x + y) dx + (x - 2 y) dy = 0$ .

Let $M = 2 x + y, N = x - 2 y$ .

Then,

$\begin{matrix} \frac{\partial M}{\partial y} = 1 \\ \frac{\partial N}{\partial x} = 1 \end{matrix}$

So, $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ .

Therefore, the given equation is exact.

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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.
$(2 x + y) dx + (x - 2 y) dy = 0$

Short Answer

Step by step solution

General form of separable, linear, exact or integrating factors

Evaluate the given equation

Testing for exactness

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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. 2x+ydx+x-2ydy=0

Short Answer

Step by step solution

General form of separable, linear, exact or integrating factors

Evaluate the given equation

Testing for exactness

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Decision Maths

Mechanics Maths

Probability and Statistics

Pure Maths

Geometry

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.
$(2 x + y) dx + (x - 2 y) dy = 0$