Question

Nonlinear ordinary differential equation

Answer 1

The coefficients of the unknown function and its derivatives in a linear differential equation are allowed to be (known) functions of the independent variable or variables; if these coefficients are constants then one speaks of a constant coefficient linear differential equation.

Answer 2

The fact about nonlinear differential equations of even second order is that there is no unifying general theory  for them. And what do you mean by  solution ? And do you consider equation which are in normal form i.e y''=f(t,y,y')  in general equations g(t,y',y'')=0 ? 1) If you want to solve the equation with subject to some initial condition y(t0)=t0 and  y'(t0)=y'0 you can of course resort to some numerical solution method. What method you should use then depends about the equation i.e is it stiff etc... 2) Do you want to know if the solution can be represented with elementary functions in closed form. Usally this can not be done and in the age of modern computers this question has almost totally lost its practical value... Series solutions are usally not very practical for nonlinear DE:s, but if you want to consider for example perturbation methods or asymptotic expansions there are special books for those too.