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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.
It happens when the solution for the equation is periodic and contains oscillatory functions such as cos, sin and their combinations.
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.