Non-linear Differential Equation - Pulling my Hair

In summary, the conversation discusses a non-linear differential equation and how to solve it using various methods such as substitution and factoring. The participants also mention a common non-linear equation and Clairaut's ODE as potential solutions. They also suggest that the equation can be simplified to y = ax² - 4a²x + b, where a and b are constants.
  • #1
jkent
2
0
Non-linear Differential Equation - Pulling my Hair !

Hi,
What seems like a simple problem could be going abit better. Any ideas would be sincerely appreciated.

(y'')^2 -xy'' + y' = 0
The squared term is causing me grief !
If I set say v = y' , that still leaves me with the squared term.
(v')^2 - xv' + v =0.

Sorry .. I must be missing something. This looks remarkably quadratic - but its not triggering anything for me right now.

Ideas please and thank you !

J. Kent.
 
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  • #2


When you let v=y', then we get:

[tex]v=xv'+f(v')[/tex]

That's a common non-linear equation. Wanna' look for it?
 
Last edited:
  • #3


Try substituting x=rcost, 4v=(rsint)^2.
 
  • #4


That's a common non-linear equation
A clue : a Clairaut's ODE
Finally:
y = a x²-4a²x+b
a , b = constants
 
  • #5


Good ideas all. I'll poke at this abit more. Claurauts ODE - hasn't made my list so far - but it clearly needs to. I was told that factoring will work. I just don't remember enough of this stuff and use it infrequently ! Thank you so much !
 

What is a non-linear differential equation?

A non-linear differential equation is an equation that involves one or more derivatives of an unknown function, with the derivatives being raised to a power or multiplied together. This makes it a non-linear equation, as opposed to a linear differential equation where the unknown function and its derivatives appear only in a linear form.

Why are non-linear differential equations difficult to solve?

Non-linear differential equations are difficult to solve because they do not have a general solution like linear differential equations. This means that each non-linear equation must be approached and solved individually, making it a more time-consuming and complex process.

What are some real-life applications of non-linear differential equations?

Non-linear differential equations are used to model a wide range of real-world phenomena, such as population growth, chemical reactions, and fluid dynamics. They are also commonly used in engineering and physics to describe complex systems and their behaviors.

What techniques are used to solve non-linear differential equations?

There are several techniques that can be used to solve non-linear differential equations, including the method of undetermined coefficients, power series solutions, and numerical methods such as Euler's method and the Runge-Kutta method. The specific technique used depends on the form and complexity of the equation.

What are some challenges in solving non-linear differential equations?

One of the main challenges in solving non-linear differential equations is that they do not have a general solution, so each equation must be approached and solved individually. Additionally, non-linear equations can exhibit chaotic behavior, making it difficult to predict the behavior of the system over time. This can make it challenging to find an accurate solution using numerical methods.

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