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

$y'=y^2$ 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.
