Dear Etiene Ubeh

We can not always get the exact solution of an ODE that why we try to construct an approximate solution. 

As Hamou mentioned,  We can not always get the exact solution of an ODE that why we try to construct an approximate solution.  Euler method is used since it is simpler, but it does not give the best solution. There are  other numerical methods better than Euler, but they might be computationally expensive.

Exact solution is not always possible  that is why we usually apply numerical methods like the Euler method, Runge kutta method etc.

We can use Euler modified or Euler improved method  to get approximate results obtained from Euler method.

As others said, we can solve differential equation if we don't have analytical solutions. Practically it would be time first derivative, so you would predict values in next time step, given initial condition.

I do not recommend this method at all. It has a low order of accuracy, and does not meet the conservation laws, which meet the differential equations. There are simple methods which do not have these disadvantages. See, for an example, the book- David Potter - Computational Physics ( read about imlicit  numerical methods)

To prove that, the approximate solution is the true or close with exact value

and simplest easy method to solve ODE and initial value problems, so the consistence are suitable for ODE.

There are many ODE can not always get a an exact solution for this we use some numerical methods such as the Euler method to prove the existence of  an approximated solution.

The Euler method is extremely simple, and can be intuitively understod by any student. The purpose of studying this method is only to show that it is possible to solve initial value problems with time stepping methods, and obtain a convergent result (although convergence is slow). It also serves to demonstrate the shortcomings of explicit time stepping methods, as one can easily demonstrate numerical instability, and analyze how numerical stability imposes a constraint on how large time steps can be used. Thus, once one has understod Euler's method, one can go on to study more advnaced methods, with faster convergence and better stability properties.