First order nonlinear differential equation Help needed.

In summary, the two equations have the same solution when u=f^{1/4} and f'=\frac{1}{f^{1/2}}. However, when u=f^{3/2} or f'=\frac{1}{f^{3/4}}, the equations have different solutions.
  • #1
plushval
4
0
Hi All,

I have been trying to solve following nonlinear differential equation, but I couldn't solve it.

0 = a*[f(t)]^{z/(z-1)} + (-t+C)*f(t) + b*[df(t)/dt]

where a, b and C are constants and 0< z<1.

Could you please help me how to solve this nonlinear differential equation? I would really appreciate any suggestions. Many thanks, in advance!

Karen
 
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  • #3
Thanks a lot for the suggestion, but unfortunately I could not make integrating factor technique work since integrating factors turns out to be a function of both "t" and "f". Any other suggestions?

Kind regards
 
  • #4
I got a suggestion: When you get stuck, fall back and work on a simpler one first, then build it back up to the original problem:

[tex]f'+f^{1/2}+(c-t)f=0[/tex]

now, what happens if I let [itex]u=f^{1/2}[/itex]

you can convert that to a DE in u right?. You know, f=u^2, f'=2uu'. You can do that? Ok, put all that back in, solve it for u, then of course f=u^2. Ok, how about:

[tex]f'+f^{3/4}+(c-t)f=0[/tex]

same dif. Let u=f^{1/4}. Now when you figure out what u is, then raise it to the fourth power to get f.

How about:

[tex]f'+f^{n/m}+(c-t)f=0[/tex]

How about when n/m is not rational?
 
Last edited:
  • #5
I will definitely try to solve it in the way that you suggested jackmell. Thanks a lot!

As a quick note, z/(z-1) is negative; hence it may cause some problems.
 
  • #6
As I said, z/(z-1) taking negative values is very problematic, therefore unfortunately, I could not solve the ODE using the approach you suggested, jackmall. I appreciate your time, though.

Any other suggestions would be highly appreciated.

Kind regards
 
  • #7
If:

[tex]f'+\frac{1}{f^{1/2}}+(c-t)f=0[/tex]

then what happens if we let:

[tex]u=f^{3/2}[/tex]

If:

[tex]f'+\frac{1}{f^{1/4}}+(c-t)f=0[/tex]

then what happens if we let:

[tex]u=f^{5/4}[/tex]
 

Related to First order nonlinear differential equation Help needed.

1. What is a first order nonlinear differential equation?

A first order nonlinear differential equation is a mathematical equation that involves an unknown function and its derivatives, where the derivative of the function is raised to a power or multiplied by another function.

2. What is the difference between linear and nonlinear differential equations?

A linear differential equation involves only the first derivative of the unknown function, whereas a nonlinear differential equation involves higher powers or products of the derivative. This makes nonlinear differential equations more complex and difficult to solve.

3. How do you solve a first order nonlinear differential equation?

Solving a first order nonlinear differential equation involves finding an expression for the unknown function that satisfies the given equation. This can be done through various methods such as separation of variables, integrating factors, or using numerical methods.

4. What are some real-life applications of first order nonlinear differential equations?

First order nonlinear differential equations have many applications in fields such as physics, engineering, economics, and biology. They can be used to model population growth, chemical reactions, motion of particles, and many other phenomena.

5. Why is it important to understand and solve first order nonlinear differential equations?

First order nonlinear differential equations are fundamental in many areas of science and engineering. They allow us to model and understand complex systems and make predictions about their behavior. Solving these equations also helps us develop new technologies and improve existing ones.

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