Solving ODE with Bernoulli's Method: y''+(y')2 = y

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SUMMARY

The discussion focuses on solving the ordinary differential equation (ODE) y'' + (y')² = y using Bernoulli's method. The user attempts a substitution with p = y' and subsequently introduces z = p² to facilitate the solution. However, confusion arises regarding the application of chain rules and the correct formulation of derivatives, particularly in the transition from p' to p. The user expresses a willingness to share their complete solution once clarified.

PREREQUISITES
  • Understanding of ordinary differential equations (ODEs)
  • Familiarity with Bernoulli's method for solving ODEs
  • Knowledge of substitution techniques in differential equations
  • Basic calculus, including differentiation and chain rule application
NEXT STEPS
  • Study the application of Bernoulli's method in detail
  • Learn about substitution techniques for solving ODEs
  • Review chain rule applications in calculus
  • Explore examples of solving ODEs with varying initial conditions
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Students and educators in mathematics, particularly those focused on differential equations, as well as anyone seeking to understand the intricacies of Bernoulli's method and its applications in solving ODEs.

manenbu
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Homework Statement



y''+(y')2 = y, y(0)=1, y'(0)=1/√2

Homework Equations



Bernoulli's method.

The Attempt at a Solution



Using the substitution p=y' I get this:
p'p + p2 = y, so I can use z=p2 to solve this.
However, I'm getting something wrong.
I think it could be because I'm having too much letters - p, z, y, x so I might be missing some chain rules.
 
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Not worked through this, but using the substitution p = y', surely you would get:

p'+p^2 = y and NOT p'p +p^2?
 
but I'm using p(y), not p(x)
so y'' = dp/dy dy/dx = dp/dy p = p' p
 
oh well, I got it. Took me 2 pages of text. If you're interested I can post the solution here. :)
 

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