Help toward solving second order non-linear differential equation

In summary, the conversation is about a second order linear differential equation with constants g and z, and the possibility of z being a function of time. The discussion also touches on the linearity of the equation and suggestions for resources to solve nonlinear differential equations.
  • #1
manikandanb
1
0
Hi,

I have a differential equation of the form
d2 x
---------------- = g/z * x(t)
dt2


Here g and z are constants. So, this is a 2nd order ODE which has a closed form solution.
In fact, i know the solution for x in terms of cosh and sinh functions.

In the above differential equation, if z is not constant. That is, id the the eqn becomes
d2 x
---------------- = g/z(t) * x(t)
dt2


Is it still correct to assume x is only a function of time?
If z(t) is sinusoidal, then this equation is a non-linear 2nd order ODE. Am i right?

Please point me to a standard textbook which might help me solve these kind of non-linear diff.equations? Any suggestions/pointers are welcome. Thank You in advance.

--MB
 
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  • #2
It's still a linear differential equation. If x and y are both solutions

[tex] \frac{d^2(x+y)}{dt^2}=\frac{d^2x}{dt^2}+\frac{d^2y}{dt^2}=\frac{g}{z(t)}x + \frac{g}{z(t)}y = \frac{g}{z(t)}(x+y)[/tex]

You can also show that scalar multiples are solutions
 

1. What is a second order non-linear differential equation?

A second order non-linear differential equation is a mathematical equation that involves the second derivative of a function, as well as non-linear terms. It can be written in the form: y'' = F(x,y,y') where y'' is the second derivative, F is a function of x, y, and y', and y' is the first derivative.

2. What is the importance of solving second order non-linear differential equations?

Second order non-linear differential equations are commonly used in fields such as physics, engineering, and economics to model real-world phenomena. Solving these equations allows us to understand and predict the behavior of these systems, making it a crucial tool in scientific research and problem-solving.

3. What are some methods for solving second order non-linear differential equations?

There are several methods for solving second order non-linear differential equations, including substitution, variation of parameters, and power series. The most commonly used method is the substitution method, where a change of variable is made to transform the equation into a first order linear differential equation, which can then be solved using standard techniques.

4. Can second order non-linear differential equations have multiple solutions?

Yes, second order non-linear differential equations can have multiple solutions. This is because there are often multiple functions that satisfy the given equation. Additionally, some equations may have a general solution that can be expressed in terms of a constant, resulting in an infinite number of solutions.

5. How can I check if my solution to a second order non-linear differential equation is correct?

To check if a solution to a second order non-linear differential equation is correct, you can substitute the solution into the equation and verify that it satisfies the equation. Additionally, you can check if the solution satisfies any initial conditions or boundary conditions given in the problem.

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