What is an nonlinear ordinary differential equation and give an example? What is an nonlinear ordinary differential equation and give an example?
And what is the difference between an linear ordinary differential equation and non-linear ordinary differential equation in formal way?
The second question seems more important to me, however, it seems too long to put in the title
 A: $y'=y^2$ is a nonlinear ODE. $y'=y$ is a linear ODE. 
In general an ODE of the form $y'=f(t,y)$ is linear if and only if $f$ is a linear function of $y$. It could have non-constant coefficients; for instance $y'=ty$ is also a linear ODE. It could also involve the components if $y$ is a vector. This means that $y''=y'+y$ is a linear ODE, because we can rewrite it as a system of two first order ODEs: $z'=z+y$ and $y'=z$. It could also involve an arbitrary function which depends only on $t$. For example, $y'=ty+t^2+\sin(t)$ is a linear ODE.
A: here is an example of a nonlinear differential equation. $$\frac{dx}{dt} = x^2, x(0) = x_0 $$ you can separate the variables and solve find $$ x = \frac{x_0}{1- tx_0}$$
you can see that there are several things different from linear equations: principle of super position does not hold, (b) the solution may not exist for all time, (c) the singularity nay depend on the initial condition. most of the nonlinear differential equation have no closed form solutions. one usually linearizes about some known state and applied the method from the linear equations.
A: In a linear ODE, the dependent function and all of its derivatives appear as linear terms.  So, if $x$ is the independent variable and $y$ the dependent one, then the general ODE is $$\sum _{n=0}^Na_n(x)y^{(n)}(x)=f(x)$$where $f(x)$ is a known function,  $y^{(0)}(x)=y(x)$, and $y^{(n)}(x)$ ($n>0$) is the $n^{th}$ derivative of $y$.  Note that all terms are linear in $y^{(n)}$
An example of a non-linear ODE is $[y'(x)]^2+y(x)=f(x)$, where $f(x)$ is a known function.  Note that the first order derivative appears as a quadratic term.  These equations are much more difficult to solve and solutions might not even exist.
