How Do You Solve a Second Order Nonlinear Autonomous ODE?

In summary, the conversation discusses a differential equation where y'' = -1/(y^2). The person is stuck and has tried guessing functions but to no avail. The other person explains that this type of equation is called an autonomous equation and can be solved by performing a simple transformation. They suggest setting y' = z and proceeding with the steps outlined.
  • #1
MHD93
93
0
Hel(lo, p)

I hope you're doing fine

I'm stuck with the following:

[itex] y'' = -1/(y^2)[/itex]

I tried guessing functions (exponentials, roots, trigs... ) , but none worked, I haven't had any DE course, so I don't have specific steps to employ,

I appreciate your help,
Thanks in advance
 
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  • #2
the case where [itex]y^{''}=f(y,y^{'})[/itex] is called an autonomous equation (x does not occur directly in the right-hand side).

It can be solved by performing a simple transformation (this transformation follows from a translational symmetry of the ODE)

Let [itex]y^{'}=z[/itex], then
[itex]y^{''}=z^{'}=\frac{dz}{dx}=\frac{dz}{dy}\frac{dy}{dx}=\frac{dz}{dy}z[/itex]
and the equation can be written as:
[itex]z\frac{dz}{dy}=-\frac{1}{y^2}[/itex]
[itex]\int z dz=-\int \frac{1}{y^2}dy[/itex],
then transform back to the original variable y and integrate again.
 

Related to How Do You Solve a Second Order Nonlinear Autonomous ODE?

1. What is a 2nd order nonlinear ODE?

A 2nd order nonlinear ordinary differential equation (ODE) is a mathematical equation that involves the second derivative of the dependent variable, as well as the dependent variable itself, in a nonlinear relationship. This means that the equation is not linear in terms of the dependent variable, and may involve terms such as powers, products, or trigonometric functions.

2. How do I solve a 2nd order nonlinear ODE?

Solving a 2nd order nonlinear ODE can be a complex process and often requires advanced mathematical techniques. The general approach involves transforming the equation into a standard form and then using methods such as substitution, variation of parameters, or power series to find a solution. In some cases, the solution may also involve numerical methods or computer simulations.

3. What are some real-world applications of 2nd order nonlinear ODEs?

2nd order nonlinear ODEs have many applications in the physical and engineering sciences, where they are used to model a wide range of nonlinear phenomena. Some examples include the motion of a pendulum, the behavior of electrical circuits, and the growth of populations in ecology.

4. Can I use software to solve a 2nd order nonlinear ODE?

Yes, there are many software programs available that can solve 2nd order nonlinear ODEs. These programs use numerical methods to approximate the solution, and they can handle a wide range of equations and boundary conditions. However, it is still important to understand the underlying mathematical concepts and techniques involved in solving these equations.

5. What are some tips for successfully solving a 2nd order nonlinear ODE?

Solving a 2nd order nonlinear ODE can be challenging, but there are some tips that can help make the process smoother. These include understanding the different types of equations and selecting the appropriate method for solving them, carefully checking for errors during the solution process, and making use of mathematical software or tools to assist with calculations.

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