In an investigation of a physics problem, I ran into the following equation: d^2(y)/(dt)^2 = k * y * (y^2 + c)^-1.5 I know how to solve separable first order differential equations but this one seems to be beyond me. Assistance?
Well, one thing you can do is multiply by y prime [tex]y^{\prime} y^{\prime \prime} = \frac{k y y^{\prime}}{(y^2 + c)^\frac{3}{2}} [/tex] and then integrate to get [tex] \frac{1}{2} y^{\prime 2} = - \frac{k}{\sqrt{y^2 + c}} + A [/tex] where A is a constant of integration. You can then square root the y prime square, pull over all the y stuff on one side (and integrate again) to get x as some horrendous integral in y. i.e., [tex]x = \int{\frac{dy}{\sqrt{2(A- \frac{k}{\sqrt{y^2 + c}})}}} [/tex] or rather [tex]x = \frac{1}{\sqrt{2}} \int{\sqrt{\frac{\sqrt{y^2 +c}}{A \sqrt{y^2 +c} - k }} dy} [/tex] Other than that, I dunno.
That looks intractable. I expected there to be a "clean" or closed (or whatever you call it) solution. This equation arose from me trying to plot the position of a point mass in a field generated by another point mass. The y is the vertical position (the reference point mass is at the origin and is stationary).