dy_ 2xy + 3y²dx x + 2xyGivenIs this a homogeneous equation? If you think so, what is your suitablereason from the followinga) No, because the degree of each term is the sameb) Yes, because the degree of numerator and denominator are not samec) Yes, because the degree of numerator and denominator is 2d) None of theseab

Answered: dy 2.xy + 3 y² Given Is this a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

dy_ 2xy + 3y²
dx x + 2xy
Given
Is this a homogeneous equation? If you think so, what is your suitable
reason from the following
a) No, because the degree of each term is the same
b) Yes, because the degree of numerator and denominator are not same
c) Yes, because the degree of numerator and denominator is 2
d) None of these
a
b

Expert Solution

Step 1

Here in the given question we have to find if the given equation is homogeneous or not and fins the reason.

Homogeneous differential equation:

$\begin{array}{l} A differential equation of the form \frac{d y}{d x} = \frac{f (x, y)}{g (x, y)} \\ is called homogeneous differential equation if \\ f (x, y) and g (x, y) are homogeneous functions \\ of the same degree in x and y . \end{array}$

Step 2

Now, check f (x, y) and g (x, y) are homogeneous function of same degree or not for the given function:

$\begin{array}{l} F (λ x, λ y) = λ^{0} F (x, y) \\ \frac{d y}{d x} = \frac{2 x y + 3 y^{2}}{x^{2} + 2 x y} or y^{'} = F (x, y) \\ F (x, y) = \frac{2 x y + 3 y^{2}}{x^{2} + 2 x y} where \{\begin{array}{l} f (x, y) = 2 x y + 3 y^{2} and \\ g (x, y) = x^{2} + 2 x y \end{array}\} \\ F (λ x, λ y) = \frac{2 (λ x) (λ y) + 3 {(λ y)}^{2}}{{(λ x)}^{2} + 2 (λ x) (λ y)} \\ = \frac{2 x y λ^{2} + 3 y^{2} λ^{2}}{x^{2} λ^{2} + 2 x y λ^{2}} = \frac{λ^{2}}{λ^{2}} \frac{2 x y + 3 y^{2}}{x^{2} + 2 x y} \\ = λ^{0} \frac{2 x y + 3 y^{2}}{x^{2} + 2 x y} \\ degree of f (x, y) = 2 x y + 3 y^{2} and g (x, y) = x^{2} + 2 x y is 2 \end{array}$

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