For your example equation, Mathematica (the system which I am using) finds no exact solution. It has, however, no problem to solve initial value and boundary value problems for this equation numerically. Don't share the wide-spread misconception that for numerical methods nonlinear problems would be particularly problematic.

One can solve non-linear equations using NRM or Mathematical Theory of Resultant and it can be also solved using A.I. Technique (i.e.- PSO, HPSO, BFO)

For your example equation, Mathematica (the system which I am using) finds no exact solution. It has, however, no problem to solve initial value and boundary value problems for this equation numerically. Don't share the wide-spread misconception that for numerical methods nonlinear problems would be particularly problematic.

You can use the ODE package. First you have to transform the second order ode in a system of two first order equations and then you can use one of the functions included in the package. For more details consult the help page of ODE. If you have no information about the stiffness of the equation use ODE45.

@Anup. I'm impressed with NRM, A.I., PSO, HPSO, BFO and always eager to learn. What is it ???

you can always solve a Non Linear Diff equation by variety of ways ..one simple is doing Taylor expansion and choosing couple of co efficients which make it linear. MATLAB has tools in Signal processing for this.

PSO ( Particle Swarm Optimisation ) or Simulated Annealing , or ANT bee colony or Genetic Algo are all couple of techniques to solve an optimisation problem.these are based on Natural phenomena with a mathematical model for the same. E.g you may have non linear control problem of a given system or plant. having done the Linear quadratic regulator optimal control we may like to design a robust system , then we can choose one of the above methods

Thx

such as grid method. such as : D2y=y(i+1)-2y(i)+y(i-1)/2h, Dy=y(i+1)-y(i)/h, y=y(i). then solve a system of nonlinear equations.

Can you try exploring the function BVP4C in Matlab, I think it might help. Cheers

It also depends on how the boundary conditions are provided.

Of course. My answer is true iff is an ODE with initial conditions. For boundary conditions the situation is very different.

In this case, one can get a solution by using Simulink, considering the x variable as the time.

Clearly, I supposed that y=y(x)

Can I solve a non-linear partial differential equation using MATLAB; when boundary conditions are given?