Solving a first order linear ODE with constant magnetic moment term

[xman]

i've derived the following differential eqn from a problem I'm working on, and i have tried in vain to solve this if anyone can give a direction where i should go our how to attack would be greatly appreciated. the eqn is

$$I\,r= -L\dot{I}+\frac{3}{2}\mu_{0}m R^{2}\frac{z \dot{z}}{\left(R^{2}+z^{2}\right)^{5/2}}$$

where $$r,L,m,\mu_{0},R,\dot{z}$$ are all constants. one of two ways I've tried solving this, was since
$$\dot{z}=const.\Rightarrow z=\dot{z}t$$
which just gives a particular solution i cannot find a solution for. the homogeneous part is quite trivial with the solution being
$$I_{homogeneous}=const. \, \exp(-rt/L)$$
am i missing something, is there another way. any help please

 

[AKG]

If L = 0, then if r = 0, I can be anything. If r is not zero, then I is just the right side divided by r. So we may assume L is nonzero. Let A = 1.5μ₀mR²(z'³L)^-1, B = R/z', D = r/L

I' + DI = At/(B² + t²)^5/2

The solution to the homogenous equation is, as you know I_H = Ce^-rt/L. So you just need a particular solution. See if this helps at all.

 

[xman]

thanks for the cite reference, i checked it out, and it's similar information i have in my old ode book in front of me. the integral you get with this
$$\int e^{Dt}\frac{At}{(B^{2}+t^{2})^{5/2}}dt$$
doesn't seem to be able to be integrated. i think this particular ode might need numerical methods to solve.