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A differential equation is defined to be an equation that contains derivatives, differentials and, of course,
the independent and dependent variable(s),or combinations thereof. Some equations may be bereft of( any )one of the above variables. Some simple examples of differential eqns. are given below:
(a) ydx xdy =0 (b) xdxydy=0 ( c )
It is easy to see that all these equations fit pretty well into the definition of differential equation. The reader can verify that the solutions to the above equations are ( respectively): y=kx, and
(Readers may substitute the solutions and observe that ,in each case,the solutions satisfy the respective ODEs).We will take a close look at a very simple equation, which is,
………………………………………………………(i)
This is a simple differential eqn. And is satisfied by the relation ………………………………(ii)
On the face of it these two equations seem to convey
nothing significant .But there is much more than meets the eye.
The first equation (i) tells us that we are talking about a curve ( or a family of curves ) where the slope of the tangent at any point (x,y) is DOUBLE THE VALUE of the x coordinate at THAT point.
Equation (ii) asserts that there is indeed a whole family of curves which satisfies the DE (i).That family consists of an infinite number of parabolas, with the Yaxis as the axis of the parabola(s).
Definition from B.S Grewal