Help with first order, Bernoulli ODE

In summary, the problem was incorrectly given and requires different substitution methods to be solved.
  • #1
scorpion990
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Help with first order, "Bernoulli" ODE

We just covered:
-First order linear ordinary differential equations
-Bernoulli Equations
-Simple substitutions.

This problem was assigned. Its supposedly a Bernoulli equation with respect to y, but I can't figure it out...

http://img520.imageshack.us/img520/12/23331767fh5.png

When I solve for dx/dy, I get dx/dy = x^3 -y/x, which is not a Bernoulli equation because of the factor of 1/x, and not x. Help?
 
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  • #3
I didn't get an answer at all. My problem is that I could not convert it into a form which I can use Bernoulli's substitutions on it. I tried finding an explicit for both y(x) and x(y).
 
  • #4
I doesn't look like Bernoulli's equation but I wonder if you can use similar techniques. Is there a change of variables you can apply to put it into a form that you know how to solve?
 
  • #5
A basic u = x^4-y substitution did not yield a linear (or a Bernoulli) differential equation =( I'm stumped. Does anybody mind steering me in the right direction?
 
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  • #6
Can you try forming the equation for the variable x?
 
  • #7
This problem was assigned. Its supposedly a Bernoulli equation with respect to y, but I can't figure it out...

The ODE [itex](x^4-y(x))\,y'(x)=x[/itex] is not a Bernoulli equation and furthermore is not a simple one :smile:
You can transformed it into a Riccati one by the transformation

[tex]x=\frac{\sqrt{t(u)}}{2^{1/6}},\,y(x)=\frac{u}{2^{2/3}},\,y'(x)=\frac{2\,\sqrt{t(u)}}{t'(u)}[/tex]

which makes the ODE

[tex]t'(u)-t(u)^2=-u[/tex]

Now letting
[tex]t(u)=-\frac{w'(u)}{w(u)}[/tex]
we arrive to

[tex]w''(u)-u\,w(u)=0[/tex]

which is the definition of the Airy function.
 
  • #8
Ya..As it turns out, our teacher gave us the wrong problem...
Thanks though! I really appreciate it!

(Its kind of interesting how tiny changes in the terms creates such a huge difference in difficulty)
 

FAQ: Help with first order, Bernoulli ODE

1. What is a first-order Bernoulli ODE?

A first-order Bernoulli ODE (ordinary differential equation) is a type of differential equation that can be written in the form dy/dx + P(x)y = Q(x)y^n, where n is a constant. It is named after the Swiss mathematician, Jacob Bernoulli.

2. How do I solve a first-order Bernoulli ODE?

To solve a first-order Bernoulli ODE, you can use the Bernoulli substitution method, where you substitute y = u^n and solve for u. Alternatively, you can use the linear substitution method, where you let v = y^(1-n) and solve for v. In both cases, you will end up with a linear differential equation that can be solved using standard methods.

3. What are the applications of first-order Bernoulli ODEs?

First-order Bernoulli ODEs have many applications in physics, engineering, and other fields. They are commonly used to model growth and decay processes, population dynamics, chemical reactions, and many other phenomena.

4. What is the difference between a first-order Bernoulli ODE and a first-order linear ODE?

The main difference between a first-order Bernoulli ODE and a first-order linear ODE is that the former has a nonlinear term (y^n) while the latter has a linear term (y). This nonlinearity makes Bernoulli ODEs more difficult to solve, but they also have a wider range of applications compared to linear ODEs.

5. Are there any real-life examples of first-order Bernoulli ODEs?

Yes, there are many real-life examples of first-order Bernoulli ODEs. One common example is the logistic equation, which models population growth with limited resources. Other examples include the radioactive decay equation, the cooling of a hot object, and the spread of infectious diseases.

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