Solving ODE Problem: Tips and Techniques for Postgrad Students

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SUMMARY

The discussion centers on solving the ordinary differential equation (ODE) given by \(\frac{dy}{dx}=\frac{y^{2}+xy^{2}}{x^{2}y-x}\). The user has attempted a substitution method but has not achieved a solution. They have transformed the equation into the form \(x(xy-1)dy=(x+1)y^{2}dx\) and are seeking further guidance on how to proceed with the separation of variables technique.

PREREQUISITES
  • Understanding of ordinary differential equations (ODEs)
  • Familiarity with separation of variables technique
  • Knowledge of substitution methods in solving ODEs
  • Basic algebraic manipulation skills
NEXT STEPS
  • Research techniques for solving separable ODEs
  • Study substitution methods specifically for nonlinear ODEs
  • Explore the use of integrating factors in ODEs
  • Learn about the existence and uniqueness theorem for ODEs
USEFUL FOR

This discussion is beneficial for postgraduate students studying mathematics, particularly those focusing on differential equations, as well as educators and tutors assisting students in advanced calculus topics.

dec46
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Hi All,

I am a postgrad student looking for help in solving this ODE.


Homework Statement



[tex]\frac{dy}{dx}[/tex]=[tex]\frac{y^{2}+xy^{2}}{x^{2}y-x}[/tex]


Homework Equations





The Attempt at a Solution


I have been attempting to solve this by substitution but without success, any help would be appreciated.

 
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It looks separable.
 
The furthest I can get is

[tex]x(xy-1)dy=(x+1)y^{2}dx[/tex]

but do not know where to go from here?
 

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