your equation may have an analytical solution obtainable through the symbolic toolbox (dsolve command).
There are many many numerical methods you can program (although I agree you will have to do a lot of work to get nearly as accurate/efficient as ode45b). The simplest (but not most accurate) is to set up the problem the same as you would for ode45: dx/xt = f(x,t) where x is a state vector. Then start with an initial condition x0. Then time step
x1 = x0 + delta-t * f(x0,t0)
x2 = x1 + delta-t * f(x1,t1)
etc
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