Integrating a differential equation

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SUMMARY

The discussion focuses on solving the differential equation y' = 2xy/(x^2 - y^2). The solution is derived using the substitution method, specifically y = vx, which simplifies the equation into a separable form. The final answer is expressed as Cy = x^2 + y^2, confirming the relationship between the variables. This method is particularly effective due to the homogeneous nature of the coefficients involved.

PREREQUISITES
  • Understanding of differential equations
  • Familiarity with substitution methods in calculus
  • Knowledge of homogeneous functions and their properties
  • Basic skills in algebraic manipulation
NEXT STEPS
  • Study the method of separation of variables in differential equations
  • Learn about homogeneous functions and their applications
  • Explore substitution techniques for solving differential equations
  • Investigate the implications of the solution Cy = x^2 + y^2 in various contexts
USEFUL FOR

Students studying calculus, particularly those focusing on differential equations, as well as educators seeking effective methods for teaching these concepts.

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Homework Statement



y' = 2xy/(x^2-y^2)

answer: Cy = x^2 + y^2

Homework Equations





The Attempt at a Solution



dy/dx = 2xy/(x^2-y^2)
dy = [2xy/(x^2-y^2)]dx

how do i separate the variables?
 
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In the form 2xy dx + (y^2-x^2) dy, the coefficients are both homogeneous and of degree two, so the substitution y=vx will work. Kind of long way to do it, but it works.
 

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