Of the following differential equations, which one is homogeneous? of the following differential equationsis homogeneous?Wich one→ (x² + y²) dx + xy dy = 0(3x4 ²-4y) dx + (x²y - x) dy = (- (³5²+4²- 5x²y) dx + (x³y ² + 4 xy) dy = 0(3xy-4) dx + (2x² + 3y²) dy=0Galaxy A71

Answered: of the following differential equations…

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

Of the following differential equations, which one is homogeneous?

of the following differential equations
is homogeneous?
Wich one
→ (x² + y²) dx + xy dy = 0
(3x4 ²-4y) dx + (x²y - x) dy = (
- (³5²+4²- 5x²y) dx + (x³y ² + 4 xy) dy = 0
(3xy-4) dx + (2x² + 3y²) dy=0
Galaxy A71

Expert Solution

Step 1: ''Introduction to the solution''

Recall the fact that a homogenous differential equations are equation that contains homogenous function.

We can solve the homogenous differential equation of the form $fraction numerator d y over denominator d x end fraction equals f left parenthesis x comma y right parenthesis$ where $f left parenthesis x comma y right parenthesis space space$ is a homogenous function by simply replacing $y equals v x rightwards double arrow fraction numerator d y over denominator d x end fraction equals v plus x fraction numerator d v over denominator d x end fraction$ .

Important fact: A function of two variables $f left parenthesis x comma y right parenthesis space$ is called homogenous function of degree $n$ if

$f left parenthesis t x comma t y right parenthesis equals t to the power of n f left parenthesis x comma y right parenthesis space space f o r space a n y space n element of straight natural numbers.$

(1) The given equation is $open parentheses x squared plus y squared close parentheses d x plus x y d y equals 0$

$rightwards double arrow fraction numerator d y over denominator d x end fraction equals negative fraction numerator open parentheses x squared plus y squared close parentheses over denominator x y end fraction equals f left parenthesis x comma y right parenthesis space left parenthesis s a y right parenthesis......... left parenthesis 1 right parenthesis$

Now, $f left parenthesis t x comma t y right parenthesis equals negative fraction numerator t squared space open parentheses x squared plus y squared close parentheses over denominator t squared x y end fraction equals space t to the power of 0 f left parenthesis x comma y right parenthesis$

$rightwards double arrow f left parenthesis x comma y right parenthesis space space$ is homogenous function of degree 0 and so the given differential equation(1) is homogenous.