I am learning how to solve differential equations (ordinary and partial)and why they are so important for physics.One thing I have noticed so far is that we know so little on the nature of the solutions of a differential equation , only very few forms of differential equations are solvable and even less forms of systems of differential equations are solvable.
I have always wondered.Small changes in the starting conditions of a system cause big changes only if the system isnt linear.Can small changes in the coefficients of the terms of a system of differential equations cause huge changes in the solution?Is there any related research going on?
Indeed, I have never used analytical solution of differential equations in practice or at least for engineering application. As you pointed out, only the simplest problems have known solutions and these are nowhere near the complex problems we face in real world.
Most of the time, engineers resort to empirical or numerical solutions. Empirical solutions come from experiments and numerical solutions are obtained through simulations such as Finite Element Method (FEM), Finite Volume Method (FVM), etc.. As my mentor told me, if we waited for analytical solutions, we wouldn't have airplanes for another couple of centuries.
As for your question, there are plenty of researches in that field e.g. sensitivity analysis, stochastic model, etc.. For example, one may study how much change in structural beam's thickness (related to coefficient of DE) affect overal structure's thickness.