Super-hard differential equation in classical mechanics problem

[snaek]

Homework Statement

A particle of mass m moves in the following (repulsive) field
U(x) = α/x², α > 0,
with α a constant parameter. Determine the (unique) trajectory of the particle, x(t), corresponding to the initial conditions of the form

x(t0) = x0 > 0,

x'(t0) = $\sqrt{\frac{2}{m}(E-\frac{α}{(x0)^2})}$

Homework Equations

^

The Attempt at a Solution

mx''(t) = -d/dx U(x)
= - (2α/x³)
= 2α/x³

=> x''(t) = 2α/mx³

Would anyone please be able to point me on the right track to solving this stupid DE? It's really starting to mess with my mind now!

 

[Chopin]

Welcome to PF!

I think case (2) on the page http://www.sosmath.com/diffeq/second/nonlineareq/nonlineareq.html corresponds to your equation, right? If so, that should help you find the answer.

 

[Chestermiller]

You can solve your "stupid" DE by multiplying both sides of the equation by x'. This will give you exact differentials on both sides.

 

[vela]

=> x''(t) = 2α/mx³

Would anyone please be able to point me on the right track to solving this stupid DE? It's really starting to mess with my mind now!

The DE always speaks well of you.

You can solve your "stupid" DE by multiplying both sides of the equation by x'. This will give you exact differentials on both sides.

This is a good trick to remember.

 

[paranormal]

I can provide some guidance for solving this differential equation. First, we can start by rewriting the equation as follows:

x''(t) = 2α/mx³

We can then use the initial conditions given to us in the problem to solve for the constant parameter α. Since we know x(t0) = x0 and x'(t0) = sqrt(2/m(E-α/(x0)^2)), we can substitute these values into the equation and solve for α. This will give us a specific value for α that we can use in our equation.

Next, we can use techniques such as separation of variables or substitution to solve the differential equation. This will involve integrating both sides of the equation to isolate x and t. It may also be helpful to graph the equation to get a visual understanding of the trajectory of the particle.

Once we have solved for x(t), we can then use this equation to determine the unique trajectory of the particle. It is important to check our solution by plugging it back into the original equation to ensure it satisfies the given initial conditions.

In addition, it may be helpful to consult with a mathematics or physics expert for further assistance in solving this differential equation. They may be able to provide more specific guidance and techniques for solving this type of problem in classical mechanics.