Non-linear differential equations

In summary, the nonlinear problem of w''(t)+2w(t)-2w3(t) = 0 can be solved by w(t) = tanh(t), as shown by replacing w'' and w with their respective values and factoring out 2tanh(t). The simplified version of the left side is 2tanh(t)(1- sech^2(t)- tanh^2(t)), with tanh^2(t)+sech^2(t) being the key to solving the problem.
  • #1
byrnesj1
7
0

Homework Statement



Show that w(t) = tanh(t) solves the nonlinear problem:

w''(t)+2w(t)-2w3(t) = 0
t ε ℝ

Homework Equations


[itex]\frac{d^2tanh(t)}{dt^2}[/itex] = -2tanh(t)sech2(t) = [itex]\frac{-8sinh(2t)cosh^2(t)}{(cosh(2t)+1)^3}[/itex]

tanh(t) = [itex]\frac{sinh(2t)}{cosh(2t)+1}[/itex]


tanh(t)3 = [itex]\frac{sinh^3(2t)}{(cosh(2t)+1)^3}[/itex]




The Attempt at a Solution


plug and chug? I'm not good at hyperbolic functions.

Any ideas?
 
Last edited:
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  • #2
yeah.. I keep getting stuck.

currently at:

[itex]\frac{[-6sinh(2t)cosh^2(t)+4cosh(2t)sinh(2t)+2sinh(2t)-2sinh^3(2t)]}{(cosh(2t)+1)^3}[/itex]
 
Last edited:
  • #3
It's very hard to decipher what you have written. Yes, if w= tanh(t) then [itex]w''= -2sech^2(t)tanh(t)[/itex] but it is not clear where you get the rest of that from. Just replacing w'' with [itex]-2sech^2(t)tanh(t)[/itex] and w with tanh(t), the left side becomes
[itex]-2sech^2(t)tanh(t)+ 2tanh(t)- 2tanh^3(t)[/itex]
Factor 2 tanh(t) out of that to get
[itex]2tanh(t)(1- sech^2(t)- tanh^2(t))[/itex]

Now, what is [itex]tanh^2(t)+ sech^2(t)[/itex]?
 
  • #4
Thank you! Also I was attempting to simplify in terms of sinh(t) and cosh(t).. I don't know why.
 

Related to Non-linear differential equations

What are non-linear differential equations?

Non-linear differential equations are mathematical equations that describe relationships between a function and its derivatives. Unlike linear differential equations, which can be solved using basic algebraic methods, non-linear differential equations require more advanced techniques such as numerical methods or approximation methods.

Why are non-linear differential equations important?

Non-linear differential equations are important because they can be used to model complex systems and phenomena in various fields such as physics, engineering, and economics. They allow scientists to understand and predict the behavior of these systems and make informed decisions for real-world applications.

What are some examples of non-linear differential equations?

Some examples of non-linear differential equations include the logistic equation, the Lotka-Volterra equations, and the Navier-Stokes equations. These equations are commonly used in population dynamics, predator-prey relationships, and fluid dynamics, respectively.

How do you solve non-linear differential equations?

Solving non-linear differential equations typically involves using numerical methods such as Euler's method, Runge-Kutta method, or the shooting method. These methods involve approximating the solution to the equation by breaking it into smaller, simpler equations and using iterative techniques to find an approximate solution.

What are the challenges of working with non-linear differential equations?

Non-linear differential equations can be challenging to work with because they do not have a general solution like linear differential equations. This means that each equation must be solved individually using specific techniques and approximations. Also, small changes in the initial conditions or parameters can lead to vastly different solutions, making it crucial to carefully analyze the results and their sensitivity to initial conditions.

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