I am having some trouble with using the shooting method : I am given a system of ode's representing initial value problem, and I know I want one of the unknowns (=u) to vanish at infinity (which I took for simplicity to be 30).

The problem is that when I am using the shooting method (I have a one-dimensional shooting. i.e.- only one of the unknowns is undetermined in (0) ), I get an oscillatory solution (the oscillations have a growing amplitude) that indeed satisfy u(30)=0, but it doesn't approach 0 uniformly (rather, it is $10^{23}$ right before 30) . The ODE becomes unstable in the direction I am shooting, so that the dominant mode completely contaminates the search for the subdominant boundary condition

Is there any suitable method for such a case ? (i.e.- I am looking for a variant of the shooting method that is able to look for a solution at infinity which is not oscillatory. rather, it should approach 0 a smooth way)

Hope I made myself clear enough.

Thanks in advance