How do I solve a 2nd order nonlinear ODE with specific boundary conditions?

In summary, a 2nd order nonlinear ODE is an equation that involves a function and its first and second derivatives, where the function is not directly proportional to its derivatives. This type of ODE differs from a 2nd order linear ODE in that the function and its derivatives can be raised to powers and multiplied together. Examples of 2nd order nonlinear ODEs include the Van der Pol oscillator, the Duffing equation, and the Lorenz system, and they are commonly used in various fields such as physics, biology, and economics. While there is no general method for solving all 2nd order nonlinear ODEs, numerical methods and approximation techniques can be used to obtain approximate solutions. These equations have many applications
  • #1
mathis314
19
0
Hi,

I need some help,

I must solve the following nonlinear differential equation,

-k1*(c'') = -k2*(c^0.5) - u*(c')

subject to the bc,

u*(c - 0.5) = k1*(c')

where k1, k2, and u are constants,

thanks
 
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  • #2
Could try Short step Fourier method (SSFM)?
 

FAQ: How do I solve a 2nd order nonlinear ODE with specific boundary conditions?

1. What is a 2nd order nonlinear ODE?

A 2nd order nonlinear ODE (ordinary differential equation) is an equation that involves a function and its first and second derivatives, where the function is not directly proportional to its derivatives. This means that the function and/or its derivatives are raised to powers or multiplied together, making the equation nonlinear.

2. How is a 2nd order nonlinear ODE different from a 2nd order linear ODE?

A 2nd order linear ODE has the form y'' + p(x)y' + q(x)y = f(x), where p(x) and q(x) are functions of x and f(x) is a function of x only. This means that the function and its derivatives are only raised to the first power and are not multiplied together. A 2nd order nonlinear ODE, on the other hand, has a more complex form with the function and/or its derivatives raised to powers and multiplied together.

3. What are some examples of 2nd order nonlinear ODEs?

Some examples of 2nd order nonlinear ODEs include the Van der Pol oscillator, the Duffing equation, and the Lorenz system. These equations arise in various fields of science and engineering, including physics, biology, and economics.

4. How are 2nd order nonlinear ODEs solved?

There is no general method for solving all 2nd order nonlinear ODEs, as the solutions can be very complex and may not have a closed-form expression. However, there are numerical methods and approximation techniques that can be used to obtain approximate solutions, and in some cases, certain types of 2nd order nonlinear ODEs can be solved analytically.

5. What are the applications of 2nd order nonlinear ODEs?

2nd order nonlinear ODEs have many applications in various fields, such as mechanics, electromagnetism, and fluid dynamics. They are used to model real-world phenomena and predict the behavior of systems. They are also important in the study of chaos and nonlinear dynamics, which have applications in fields such as weather forecasting and population dynamics.

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