Analyzing a Particle in a Non-Homogeneous Differential Equation

In summary, Sandra is trying to solve a problem involving a conservative force and the potential, but she is having trouble getting started. She recommends starting with the incorrect equation and then trying to solve for the derivatives.
  • #1
SandraH
5
0

Homework Statement







We're required to analyse a particle moving in the potential U(x) = a/x^2 (a > 0). Setting F = U(x) and using the Newton equation F = ma, this gives rise to the DE:



d2x = a/m * (1/x^2)

dt^2



I can't for the life of me figure out what method to use or even what sort of DE this is! That 1/x^2 is blowing my mind.







Any clues would really help.


The Attempt at a Solution




I've tried treating this as a non-homogeneous constant-coefficient ODE of the form



ax'' + bx' + cx = f(t)



(setting f(t) = 1/ x(t)^2 )



I try to solve



d2x = x(t)^-2

dt^2



Ignoring constants for now.



Complementary solution (solution to the complementary equation:
ax"(t) + bx'(t) + cx(t) + d = 0)



-> x_c = at + b



Particular solution



-> xp = x(t)^-2 + x(t)^-1 + x(t) + c1 ?



Next I find the derivatives of the particular solution





x'p = -2x(t)^-3 * x'(t) -x(t)^-2 * x'(t) + x'(t)



And using the chain rule:



x''p = [ -2x(t)^-3 * x''(t) + x'(t) * 6 x(t)^-3 * x'(t)] + [-x(t)^-2 * x''(t) + x'(t)* 2x(t)^-3 * x'(t)] + x''(t)



At this point (right before finding constants) I seriously begin to doubt whether this is the method to use. Any hints?









I also tried using the modified Newton equation (t - t0 = +/- sqrt(m/2) int[1/sqrt(E-U(y)] dy... it's called 'reduction to quadrature') which yielded a very hard integral I couldn't do by substitution or parts



int ((E - a/y^2)^-1/2 ) dy



I'm yet to try partial fractions, but I seriously doubt whether that would work...



Thanks!



Sandra
 
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  • #2
Welcome to PF Sandra,

Thanks for posting such a complete post detailing your efforts, it's refreshing change from students who just expect us to do their homework for them! Your problem is right at the beginning of your solution:
SandraH said:
We're required to analyse a particle moving in the potential U(x) = a/x^2 (a > 0). Setting F = U(x)
This is incorrect, the relationship between a conservative force F and the potential U is,

[tex]\underline{F} = -\nabla U[/tex]

Which in one dimension simplifies to,

[tex]F = -\frac{d}{dx}U[/tex]
 
  • #3
ahaaa a problem right at the beginning. I feel sheepish..
Thanks Hootenanny
 
  • #4
SandraH said:
ahaaa a problem right at the beginning. I feel sheepish..
Thanks Hootenanny
Your welcome :smile:
 

1. What is a differential equation?

A differential equation is an equation that relates an unknown function to its derivatives. It is commonly used in mathematics, physics, and engineering to model real-world phenomena.

2. Why are differential equations important?

Differential equations are important because they allow us to describe and predict the behavior of complex systems in various fields, such as physics, biology, economics, and engineering. They also provide a powerful tool for solving many practical problems.

3. What are the different types of differential equations?

There are several types of differential equations, including ordinary differential equations, partial differential equations, and stochastic differential equations. Ordinary differential equations involve one independent variable, while partial differential equations involve multiple independent variables. Stochastic differential equations deal with random processes.

4. How do you solve a differential equation?

The method for solving a differential equation depends on its type and complexity. Some equations can be solved analytically, while others require numerical methods. Common techniques for solving differential equations include separation of variables, variation of parameters, and Laplace transforms.

5. What are some real-world applications of differential equations?

Differential equations are used to model and analyze a wide range of real-world phenomena, including population growth, chemical reactions, heat transfer, fluid dynamics, and electrical circuits. They are also used in the development of many technologies, such as airplanes, cars, and computer simulations.

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