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

In summary, the problem involves finding a solution for the differential equation y''+(y')^2 = y, with initial conditions y(0) = 1 and y'(0) = 1/√2. Bernoulli's method is used, and the substitution p = y' is made, resulting in p'p + p^2 = y. However, a mistake may have been made due to the number of variables involved. After realizing this, the correct substitution is used (p = y'(x)) and the solution is found after some extensive calculations.
  • #1
manenbu
103
0

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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  • #2
Not worked through this, but using the substitution p = y', surely you would get:

p'+p^2 = y and NOT p'p +p^2?
 
  • #3
but I'm using p(y), not p(x)
so y'' = dp/dy dy/dx = dp/dy p = p' p
 
  • #4
oh well, I got it. Took me 2 pages of text. If you're interested I can post the solution here. :)
 

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

1. What is Bernoulli's Method?

Bernoulli's Method is a technique used to solve ordinary differential equations (ODEs) that can be reduced to a specific form: y'' + p(x)y' + q(x)y = r(x)y^n, where n is any real number other than 0 or 1. This method involves substituting a new variable, v, for the original variable, y, and manipulating the equation to turn it into a linear differential equation that can be solved.

2. How does Bernoulli's Method work?

To solve an ODE using Bernoulli's Method, the first step is to identify the equation as a Bernoulli equation and then substitute v = y^(1-n) for the original variable y. Next, we use the chain rule to rewrite the equation in terms of v and solve it as a linear differential equation. Finally, we substitute the solution back into the original equation to obtain the solution for y.

3. What are the advantages of using Bernoulli's Method?

Bernoulli's Method is particularly useful for solving nonlinear ODEs, as it allows us to reduce them to linear equations that are generally easier to solve. It also provides a systematic approach to solving ODEs and can be used to solve a wide range of problems in various fields of science and engineering.

4. What are the limitations of Bernoulli's Method?

Bernoulli's Method can only be used for ODEs that can be reduced to a specific form: y'' + p(x)y' + q(x)y = r(x)y^n. This means that it cannot be applied to all ODEs. Additionally, the method can be quite tedious and time-consuming, especially for higher-order equations or equations with complicated coefficients.

5. Can Bernoulli's Method be applied to initial value problems?

Yes, Bernoulli's Method can be used to solve initial value problems. However, the initial conditions must be given in terms of the original variable y, not the substituted variable v. Once the solution is obtained, we can use the initial conditions to determine the values of the arbitrary constants and obtain the final solution.

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