Some partial differential equations do not have analytical solutions. These can only be solved numerically.
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Very often because no analytical solution is available.
The answer will depend very much on the nature of the equation. The steps required for a one-step equation are very different from the steps required for a partial differential equation. For some equations there are no straightforward analytical methods of solution: only numerical methods.
ordinary differential equation is obtained only one independent variable and partial differential equation is obtained more than one variable.
Yes, it is.
All the optimization problems in Computer Science have a predecessor analogue in continuous domain and they are generally expressed in the form of either functional differential equation or partial differential equation. A classic example is the Hamiltonian Jacobi Bellman equation which is the precursor of Bellman Ford algorithm in CS.