Solving a Non-Linear Equation: Need Help!

[CPL.Luke]

hey does anybody have any idea how to solve this equation?

E/m= xd^2x/dt^2with initial conditions x=a and dx/dt(a)=0

its non-linear and so I don't have any idea what to do with it, and maple won't give me an answer.

 

[Matthew Rodman]

Yeah, it's a tricky Emden-Fowler equation. The only thing I can think of is to multiply by x' and integrate to get

$$\frac{1}{2}x^{\prime 2} = \frac{E}{m} \ln{x} + C$$

Now, swap for x from the original equation, i.e.,

$$x = \frac{E}{m x^{\prime \prime}}$$

such that

$$\frac{1}{2}x^{\prime 2} = \frac{E}{m} \ln{\frac{E}{m x^{\prime \prime}}} + C$$

rearrange to get

$$x^{\prime \prime} = A e^{-\frac{m x^{\prime 2}}{2 E}}$$

where A is a constant. Now make the change of variable

$$x^{\prime} = i \sqrt{\frac{2 E}{m}} z$$

to get

$$i \sqrt{\frac{2E}{m}} z^{\prime} e^{-z^2} = A$$

but of course

$$\frac{ d }{dt} erf{(z)} = \sqrt{\frac{2}{\pi}} z^{\prime} e^{-z^2}$$

so integrating gives you

$$erf{(z)} = \alpha t + \beta$$

where alpha and beta are constants that may or may not be complex

So there's a solution of the form

$$x(t) = i\sqrt{\frac{2 E}{m}} \int{erf^{-1}{(\alpha t + \beta)} dt}$$

Note: the inverse erf function is integrable -- see this Wolfram page.

 

[tacman]

Hello! Solving non-linear equations can be tricky, but there are a few methods you can try. One approach is to use numerical methods, such as the Newton-Raphson method or the secant method, to approximate a solution. Another option is to use a graphing calculator or software to plot the equation and visually identify any potential solutions.

Additionally, you can try to simplify the equation by using substitution or rearranging terms. It may also be helpful to review any relevant mathematical concepts or equations that could potentially assist in solving the problem.

I understand that Maple is not giving you an answer, but it may be worth trying a different software or asking a math tutor or instructor for assistance. I hope this helps and good luck with solving your equation!