Differential Eq using Substitution help

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SUMMARY

The discussion focuses on solving the differential equation (x^2)y' + 2xy = 5y^4. The user simplifies the equation to y' = [(5y^4)/(x^2)] - 2y/x and attempts substitution with v = y/x. However, the conversation highlights that the equation is a Bernoulli type, suggesting the substitution v = 1/y^3 as a more effective approach. The importance of recognizing the type of differential equation for selecting appropriate solving techniques is emphasized.

PREREQUISITES
  • Understanding of differential equations, specifically Bernoulli equations
  • Familiarity with substitution methods in solving differential equations
  • Knowledge of derivatives and their notation
  • Basic algebraic manipulation skills
NEXT STEPS
  • Research Bernoulli differential equations and their characteristic substitutions
  • Practice solving differential equations using various substitution methods
  • Explore the implications of different types of differential equations on solution strategies
  • Study the application of the substitution v = 1/y^n in solving non-linear equations
USEFUL FOR

Students and educators in mathematics, particularly those studying differential equations, as well as anyone seeking to improve their problem-solving techniques in this area.

g.sharm89
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Hey. I am having a hard time solving this problem.


(x^2)y' + 2xy = 5y^4

I get as far as simplifying to
y' = [(5y^4)/(x^2)] - 2y/x

Then use v: y/x and y: vx & y': v'x + v

And get

v'x + v = [5(v^4)(x^2)] - 2v


And then I get lost. Any help would be appreciated. Thanks!
 
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The convenient substitution is v=1/y^3
 
Agreed. while [tex]v = y/x[/tex] does work quite frequently (at least in your homework problems I imagine), you have probably learned about other trademark substitutions of different types of equations. Specifically, this is a Bernoulli type equation (and, it has its own characteristic substitution, and JJacquelin has informed which specific substitution to use in this problem). Learning to diagnose a differential equation and determine what type it is, is a quick way to know immediately how to solve them. I encourage you to look up Bernoulli differential equations online, you will find a lot of material.
 

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