Solving a non-linear non-seperable differential equation

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SUMMARY

The discussion focuses on solving the non-linear non-separable differential equation (2x^2)yy' = −1. The equation is transformed into a separable form, 2y dy = -dx/x^2, which simplifies the solving process. The participant initially struggles with the separability of the equation but realizes the solution is straightforward upon further examination. The exchange highlights the importance of recognizing transformation techniques in differential equations.

PREREQUISITES
  • Understanding of differential equations, specifically non-linear types.
  • Familiarity with the concept of separability in differential equations.
  • Knowledge of basic calculus, including differentiation and integration.
  • Experience with algebraic manipulation of equations.
NEXT STEPS
  • Study methods for solving non-linear differential equations.
  • Learn about the technique of variable separation in differential equations.
  • Explore integration techniques for solving differential equations.
  • Investigate the application of differential equations in real-world scenarios.
USEFUL FOR

Students, mathematicians, and engineers interested in solving complex differential equations and enhancing their problem-solving skills in mathematical analysis.

SuperNomad
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I'm having trouble solving this differential equation:

(2x^2)yy' = −1

I'm just really not sure how to go about solving this.
 
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Why isn't it separable?

2x^2 y \frac{dy}{dx} = -1 \Rightarrow 2y dy = -\frac{dx}{x^2}
 
Ah yes, that's actually quite easy, I'm just being dense.

Thanks for the help.

SuperNomad
 

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