
The
GENERAL SOLUTION of a D.E. is the set of all
of it's particular soltions, often expressed using a constant C
(or K) which could have any fixed value. 

As examples, y = 2x +C is the general solution of example [a] above,
and y = x^{3}  4x + C is the general solution of example [b] above. 

If (as in [b] above), y' = 3x^{2}  4
and [y = 4] when [x = 1], then first antidifferentiate [b]
to obtain a GENERAL SOLUTION : y = x^{3}  4x + C [as shown in the right figure below].
Then substitute x = 1 and y = 4, to get : 4 = 1^{3}  4(1) + C.
Solve for C to get C = 1.
Therefore, the PARTICULAR SOLUTION
for which [y = 4] and [x = 1] is [y = x^{3}  4x  1]
Use the graph below to display this particular solution, which
must pass through the point (1, 4). 

The collection of graphs of all particular solutions of a D.E. completely
fill up space, and no two graphs overlap, much as the layers of a dagwood sandwich. The red graphs in the
right image are playing the same role as the ham, cheese, and lettuce in Dagwood's sandwich in the left image. 



