Homogenous differential equation

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SUMMARY

The discussion focuses on solving the homogeneous differential equation dy/dx = (y^2 + x*sqrt(x^2 + y^2))/xy. The key method involves separating variables into the form of y/x, allowing the substitution y = ux, where u is a function of x. The participant expresses difficulty in transforming the equation into the appropriate form for simplification and solution.

PREREQUISITES
  • Understanding of homogeneous differential equations
  • Familiarity with variable separation techniques
  • Knowledge of substitution methods in differential equations
  • Basic calculus concepts, including derivatives and functions
NEXT STEPS
  • Study the method of substitution in homogeneous differential equations
  • Learn about variable separation techniques in differential equations
  • Explore examples of solving differential equations using the form y = ux
  • Review the properties of derivatives and their applications in differential equations
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Students studying differential equations, mathematics educators, and anyone seeking to enhance their problem-solving skills in homogeneous differential equations.

rbailey5
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1. Homework Statement [/b

so I am given a known homogeneous differential equation dy/dx=(y^2+x*sqrt(x^2+y^2))/xy


Homework Equations


now I know that you have to separate into some form of y/x which then you can change into v and solve the differential equation but I am having trouble


The Attempt at a Solution


I just can't figure out how to get it in the appropriate form in order to simplify
 
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Let y = ux, y' = u'x + u. Work it out and show us what happens.
 

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