Chapter 2: Q3E (page 64)

In problems $1 - 8$ Classify the equation as separable, linear, exact, or none of these. Notice that some equations may have more than one classification.
$\sqrt{- 2 y - y^{2}} dx + (3 + 2 x - x^{2}) dy = 0$

Short Answer

Expert verified

The equation is separable.

Step by step solution

Find if the equation is separable

Here

$\sqrt{- 2 y - y^{2}} dx + (3 + 2 x - x^{2}) dy = 0 \frac{dy}{dx} = \frac{\sqrt{- 2 y - y^{2}}}{x^{2} - 2 x - 3}$

This equation can be written in the form as g(x) h(y).so, this equation is separable.

Determine if the equation is linear

This equation is not linear of y and x because the RHS is not linear with respect to y and x respectively.

Evaluate whether the equation is exact

The condition for exact is $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ .

$M (x, y) = \sqrt{- 2 y - y^{2}} N (x, y) = 3 + 2 x - x^{2}$

$\frac{\partial M}{\partial y} = \frac{- 1 - y}{\sqrt{- 2 y - y^{2}}} \frac{\partial N}{\partial x} = 2 - 2 x \frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$

This equation is not exact.

Hence, the equation is separable.

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In problems $1 - 8$ Classify the equation as separable, linear, exact, or none of these. Notice that some equations may have more than one classification.
$\sqrt{- 2 y - y^{2}} dx + (3 + 2 x - x^{2}) dy = 0$

Short Answer

Step by step solution

Find if the equation is separable

Determine if the equation is linear

Evaluate whether the equation is exact

One App. One Place for Learning.

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In problems 1-8Classify the equation as separable, linear, exact, or none of these. Notice that some equations may have more than one classification.-2y-y2dx+(3+2x-x2)dy=0

Short Answer

Step by step solution

Find if the equation is separable

Determine if the equation is linear

Evaluate whether the equation is exact

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Mechanics Maths

Probability and Statistics

Applied Mathematics

Decision Maths

Calculus

Study anywhere. Anytime. Across all devices.

In problems $1 - 8$ Classify the equation as separable, linear, exact, or none of these. Notice that some equations may have more than one classification.
$\sqrt{- 2 y - y^{2}} dx + (3 + 2 x - x^{2}) dy = 0$