Q: What is oscillatory solution in differential equations?

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Very often because no analytical solution is available.

Some partial differential equations do not have analytical solutions. These can only be solved numerically.

Euler's Method (see related link) can diverge from the real solution if the step size is chosen badly, or for certain types of differential equations.

Finite Differential Methods (FDM) are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.

No, analytical solutions do not always exist. That is to say, the answer need not be a function. However, it is possible to find numerical solutions.

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Very often because no analytical solution is available.

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