Question

example of ordinary differential equation and partial differential equation.

Answer 1

All-ready answered

Answer 2

Differential Equations are of two types, ODE and PDE as you asked an example of each.
but there is classification of each type, for example, in ODE, an ODE equation is classified by its linearity, order, being homogeneous or non homogeneous... and so goes for PDE.

better if you refer to a text on ODE first (PDE's are changed to ODE before solving)...

just an example of Linear First Order Ordinary Differential Equation
\[dy/dx + p(x)y=0\]
where p(x) is the coefficient, function of x.

there are many terminologies involved ....