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Tuesday, 5 August 2014

CHAPTER 2 - Basic Concepts and Terminology

These three examples should be sufficient for you to realise why and how differential equations arise and why they are important. In all the three equations mentioned above, there is only independent variable (the time t in all the three cases). Such equations are termed ordinary differential equations. We might have equations involving more than one independent variable:
\dfrac{{\partial f}}{{\partial x}} + x\dfrac{{\partial f}}{{\partial y}} = {x^2}
where the notation \dfrac{\partial }{{\partial x}} stands for the partial derivative, i.e., the term \dfrac{{\partial f}}{{\partial x}} would imply that we differentiate the function f with respect to the independent variable x as the variable (while treating the other independent variable y as a constant). A similar interpretation can be attached to \dfrac{\partial }{{\partial y}}.
Such equations are termed partial differential equations but we’ll not be concerned with them in this chapter. Consider the ordinary differential equation
\dfrac{{{d^2}y}}{{d{x^2}}} + x\dfrac{{dy}}{{dx}} + {x^2} = c
The order of the highest derivative present in this equation is two; thus, we’ll call it a second order differential equation (DE, for convenience).
The order of a DE is the order of the highest derivative that occurs in the equation
Again, consider the DE
{\left( {\dfrac{{{d^3}y}}{{d{x^3}}}} \right)^2} + \dfrac{{dy}}{{dx}} = {x^2}{y^2}
The degree of the highest order derivative in this DE is two, so this is a DE of degree two (and order three).
The degree of a DE is the degree of the highest order derivative that occurs in the equation, when all the derivatives in the equation are made of free of fractional powers
For example, the DE
\sqrt {{{\left( {\dfrac{{dy}}{{dx}}} \right)}^2} - 1}  + x{\left( {\dfrac{{{d^2}y}}{{d{x^2}}}} \right)^2} = k
is not of degree two. When we make this equation free of fractional powers, by the following rearrangement,
{\left( {\dfrac{{dy}}{{dx}}} \right)^2} - 1 = {\left\{ {k - x{{\left( {\dfrac{{{d^2}y}}{{d{x^2}}}} \right)}^2}} \right\}^2}
we see that the degree of the highest order derivative will become four. Thus, this is a DE of degree four (and order two).
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