See this,

http://math.stackexchange.com/questions/125702/the-number-of-solutions-to-an-nth-order-differential-equation

http://math.stackexchange.com/questions/1089286/an-nth-order-ode-has-n-linearly-independent-solutions

Read Linear Algebra, Basis and Vector space concept. You get a good idea in this

http://booksite.elsevier.com/samplechapters/9780123820105/Front_Matter.pdf

You could find the detailed answer in this book (see chapter 5).

I believe that you are speaking about LINEAR homogeneous differential equations. Am I right?

The result is based on the theorem that the initial value (Cauchy ) problem for linear differential equation has unique solution. In other words, if you have an equation of n-th order and a point t0, then for a given vector (a0, a1, ..., an-1) there is a unique solution f of the equation that satisfies f(t0) = a0, f ' (t0) = a1, ... , f(n-1)(t0) = an-1. This theorem means, that there is a bijective linear operator which maps each vector (a0, a1, ..., an-1) to the corresponding solution f. Since the dimension  of the space of vectors (a0, a1, ..., an-1) equals n, the dimension  of the space of solutions also equals n.

Of course, the statement of the above theorem is incomplete: some details about the domain of coefficient functions, their properties etc. are missing. These details can be found in every decent  textbook on differential equations.

@ Kadets

sir can you name any book

I referred WIley-Shepley Ross but it doesnot cover that proof.

The textbooks  that I used when I was a student are in Russian, which I afraid make them of little use for you :).  I have made a google search with keyword ``existence and uniqueness of solution for linear differential equation'' and have found immediately the following text that contains the exact statement of the theorem and the reference  to a texbook:

https://math.berkeley.edu/~shinms/SP14-54/diffeq-thms.pdf

It also contains the explanation, why in the homogeneous case there are n linearly independent solutions, that I sketched in my previous post, and what happens in the non- homogeneous case.

Another reference:

D. Willett.  The Existence-Uniqueness Theorem for an nth Order Linear Ordinary Differential Equation.  The American Mathematical Monthly

Vol. 75, No. 2 (Feb., 1968), pp. 174-178 

https://www.jstor.org/stable/2315901?seq=1#page_scan_tab_contents

See the clásico book of Earl Coddington,  theory of ordinary differential equations...or linear ordinary differential equations by earl Coddington and Robert carlson. Regards

This result of N independent solutions is valid for regular linear differential equations. The characteristic equation for it (see https://en.wikipedia.org/wiki/Characteristic_equation_(calculus)  )  represents a polynomial of N-th order, which always has N roots (complex!) by the main theorem of algebra.

For nonlinear differential equations this might be not true. For equations with variable coefficients there might be critical points. There is also a phenomenon of bifurcation in critical points; see https://en.wikipedia.org/wiki/Bifurcation_theory for the beginning.

The best way to prove that n solutions to a linear n-th order differential equation spans all of the solutions makes use of the Wronskian determinant, defined as the determinant of the matrix with columns are given by the n solutions themselves. Based on the Abel's identity for the Wronskian, linear dependency/independency of the n solutions at a point is preserved at any point giving as a consequence of its exponential-type form. Uniqueness for the Cauchy problem (and smooth dependency with respect to the unknown) does the rest of the job.

You can see this

http://math.stackexchange.com/questions/1089286/an-nth-order-ode-has-n-linearly-independent-solutions