Why are these DEs homogeneous?

  • Thread starter Thread starter bcjochim07
  • Start date Start date
  • Tags Tags
    Homogeneous
Click For Summary
SUMMARY

The discussion focuses on the classification of differential equations (DEs) as homogeneous, specifically addressing the equations (x + ye^(y/x))dx - (xe^(y/x))dy = 0 and ydx + x(lnx - lny - 1)dy = 0. Participants express confusion regarding the concept of homogeneity, particularly in relation to the presence of exponents and the substitution methods y=ux or v=xy. The key takeaway is that a DE is considered homogeneous if all terms can be expressed as a function of the same degree, allowing for substitution techniques to simplify the equations.

PREREQUISITES
  • Understanding of homogeneous differential equations
  • Familiarity with substitution methods in differential equations
  • Knowledge of exponential functions and logarithms
  • Basic skills in solving first-order differential equations
NEXT STEPS
  • Study the theory behind homogeneous differential equations
  • Practice substitution methods, specifically y=ux and v=xy
  • Explore examples of solving homogeneous DEs with varying degrees
  • Learn about the implications of exponents in differential equations
USEFUL FOR

Students and educators in mathematics, particularly those focusing on differential equations, as well as anyone looking to deepen their understanding of homogeneity in DEs.

bcjochim07
Messages
366
Reaction score
0

Homework Statement


The following DEs are homogeneous and can be solved using a substitution y=ux or v=xy. I can solve them, I'm just don't see how they are homogeneous.

How is it that with there being an exponent (y/x) that this is homogeneous. I don't see how I could pull out t[tex]\alpha[/tex]

(x + ye^(y/x))dx - (xe^(y/x))dy = 0

-------------------------------
Same problem with this one

ydx + x(lnx - lny -1)dy=0

For both, I can see the powers are the same, but I don't think I completely understand what it means to have a homogeneous differential equation.




Homework Equations





The Attempt at a Solution

 
Physics news on Phys.org
also this one is bothering me x(dy/dx) = y ln(xy)
 

Similar threads

Show that ODE is homogeneous, but I don't think it is
  • · Replies 2 ·
Replies
2
Views
2K
Solving system of differential equations using elimination method
  • · Replies 10 ·
Replies
10
Views
2K
Homogenous solution of a differential equation
  • · Replies 10 ·
Replies
10
Views
2K
Solve (t^2-1)y'' +4ty'+2y=6t, given two particular solutions
  • · Replies 2 ·
Replies
2
Views
1K
Solve ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##
  • · Replies 7 ·
Replies
7
Views
2K
Homogeneous Differential Equation
  • · Replies 7 ·
Replies
7
Views
3K
Homogeneous Diff. Eqn Finding Solution
  • · Replies 5 ·
Replies
5
Views
2K
Solving a Non-Homogeneous Linear Differential Equation
  • · Replies 12 ·
Replies
12
Views
3K
Substitution to convert first order ODE to homogenous
  • · Replies 1 ·
Replies
1
Views
1K
ODE with constant coefficient ##b##
  • · Replies 11 ·
Replies
11
Views
2K