What kind of problem is this. (Seperable or Bernoullis) / Diff EQ

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SUMMARY

The discussion focuses on solving the initial value problem defined by the differential equation xy² dy/dx = y³ - x³ with the initial condition y(1) = 2. Participants clarify that the equation is not a Bernoulli type and explore methods for separation of variables. A suggested substitution is y = xv, which can simplify the equation for further analysis. The correct separation of variables is confirmed as y² - y³ dy = -x² dx.

PREREQUISITES
  • Understanding of first-order differential equations
  • Familiarity with Bernoulli differential equations
  • Knowledge of separation of variables technique
  • Basic skills in substitution methods for solving differential equations
NEXT STEPS
  • Research the method of substitution in differential equations, specifically y = xv
  • Study the characteristics of Bernoulli differential equations
  • Learn about the separation of variables technique in depth
  • Explore initial value problems and their solutions in differential equations
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Students studying differential equations, educators teaching calculus, and anyone seeking to improve their problem-solving skills in mathematical analysis.

Squizzel
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Homework Statement



xy^2 dy/dx = y^3 - x^3 , y(1) = 2

Homework Equations


The Attempt at a Solution



It says to solve the initial value problem. I am assuming it is not a Bernoulli, but I can't seem to separate it. What should I do?Thanks

This is what I get when I separate it, is this right?y^2-y^3 dy = -x^2 dx
 
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Squizzel said:

Homework Statement



xy^2 dy/dx = y^3 - x^3 , y(1) = 2


Homework Equations





The Attempt at a Solution



It says to solve the initial value problem. I am assuming it is not a Bernoulli, but I can't seem to separate it. What should I do?


Thanks

This is what I get when I separate it, is this right?


y^2-y^3 dy = -x^2 dx

xy^2 dy/dx = y^3 - x^3
dy/dx = y/x - x^2/y^2
 
Squizzel said:

Homework Statement



xy^2 dy/dx = y^3 - x^3 , y(1) = 2

Homework Equations



The Attempt at a Solution



It says to solve the initial value problem. I am assuming it is not a Bernoulli, but I can't seem to separate it. What should I do?

Thanks

This is what I get when I separate it, is this right?

y^2-y^3 dy = -x^2 dx
How do you get that last line?

It should be y2 dx - y3 dy = -x2 dx, which is not separated.

Try the substitution, y = xv .
 

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