Turing solvable means the problem can be solve in recursive function, and vice versa. Any problem that can be solved with finite steps with finite resource is Turing solvable. The example you gave, any differential equation that has solution, regardless whether it is numeric or analytical, is Turing solvable. However, there set of all differential equations is non-recursively enumerable, hence there is no way to solve them all, you can't even count them.