Second-order nonlinear ordinary differential equation

In summary: Hi lanedance, I tried that method before and got (1/2)*(u^2)=1/(x^2)(try using the X2 tag just above the Reply box :wink:)
  • #1
lewis198
96
0

Homework Statement



Given the Second-order nonlinear ordinary differential equation

x''(t)=1/(x(t)^2)

Find x(t).

Homework Equations



I tried use Laplace transforms, and solving it using linear methods but that is not useful.

The Attempt at a Solution



I tried to find t(x) and got to dt=dx/((C-2GM/x)^0.5) or something like that.

I guess you could find t(x) then find [inverse t(x)] = x(t)
But I would like to know how to solve it properly really.
 
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  • #2
lewis198 said:
x''(t)=1/(x(t)^2)

Find x(t).

Hi lewis198! :smile:

Standard trick: multiply both sides by x'(t) :wink:
 
  • #3
similar outcome is to susbtitute to get a seprable equation then intgerate twice (ash means derivative w.r.t. t)
u = x'
then
x" = u' = (du/dx)(dx/dt) = (du/dx)
 
  • #4
Hi lanedance, I tried that method before and got

(1/2)*(u^2)=1/(x^2)

dt=dx/((C-2GM/x)^0.5).

this will therefore give me an integral t(x).

But I need x(t). It will be quite messy doing the inverse won't it?

I'm not sure where to go from multiplying LHS and RHS by x'(t).

Is there a more elegant way to get x(t)? For example if I had

ax''+bx'+cx=f(x)

I could get y=A*e^(mt)+B*e(mt)

But since in my characteristic equation b=0 and c=0 my m quadratic equation is void.
 
  • #5
I'll just find t(x) then t-1(x)
 
  • #6
lewis198 said:
x''(t)=1/(x(t)^2)

lewis198 said:
Hi lanedance, I tried that method before and got

(1/2)*(u^2)=1/(x^2)

(try using the X2 tag just above the Reply box :wink:)

No, the RHS is wrong and you've left out the constant of integration. :wink:
 
  • #7
yeah I'm not too sure, don't know if Tim has any other ideas, but simple generic general solutions don't always apply to non-linear de's - this one is gets a little crazy near x=0, and tends to a straight line for x>>1

So the general solution may not be able to be solved simply for t^(-1). That said if you have the right boundary conditions, this one could simplify a bit... (in particular if you could set the 1st constant of integration to zero)
 
Last edited:
  • #8
lanedance said:
similar outcome is to susbtitute to get a seprable equation then intgerate twice (ash means derivative w.r.t. t)

ash?
 
  • #9
Mark44 said:
ash?
there's a fire... few mistakes, so i clarified below
--------------------------------------------------------------------------------

similar outcome is to susbtitute to get a separable equation then integrate twice (dash means derivative w.r.t. t)
[tex] u = x' = \frac{dx}{dt} [/tex]
then
[tex] x'' = u' = \frac{du}{dt} =\frac{du}{dx} \frac{dx}{dt} = \frac{du}{dx}u = \frac{1}{x^2} [/tex]
 

1. What is a second-order nonlinear ordinary differential equation?

A second-order nonlinear ordinary differential equation is a mathematical equation that involves a dependent variable, its derivatives, and possibly the independent variable, where the highest order derivative is squared or multiplied by the dependent variable. It is a type of differential equation that is commonly used in physics, engineering, and other scientific fields to model complex systems.

2. How is a second-order nonlinear ordinary differential equation different from a first-order equation?

The main difference between a second-order nonlinear ordinary differential equation and a first-order equation is that the former involves the square or multiplication of the dependent variable's highest order derivative, while the latter only involves the derivative itself. This means that second-order equations are more complex and generally more difficult to solve analytically.

3. What are some examples of second-order nonlinear ordinary differential equations?

Some common examples of second-order nonlinear ordinary differential equations include the damped harmonic oscillator equation, the Lotka-Volterra predator-prey model, and the Van der Pol oscillator equation. These equations are used to model various physical phenomena such as oscillations, population dynamics, and electrical circuits.

4. How are second-order nonlinear ordinary differential equations solved?

Solving a second-order nonlinear ordinary differential equation involves finding a function that satisfies the equation. This can be done analytically using techniques such as separation of variables, substitution, or series solutions. In some cases, numerical methods may also be used to approximate a solution.

5. What are the applications of second-order nonlinear ordinary differential equations?

Second-order nonlinear ordinary differential equations have a wide range of applications in various fields such as physics, engineering, biology, and economics. They can be used to model complex systems and phenomena such as vibrations, chemical reactions, population dynamics, and electrical circuits. Understanding and solving these equations can provide valuable insights into the behavior of these systems and aid in making predictions and decisions.

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