DIFFERENTIAL EQUATIONS

 Definition 1
A DIFFERENTIAL EQUATION (D.E.) is an equation which "contains" derivatives of some function.

EXAMPLES :
 [a] y' = 2 [b] y' = 3x2 - 4 [c] y' = y2 [d] y' = y' [e] (y' )2 + y2 = 1
 Note that the word "contains" is not clearly defined, so that in the 4th DE, y' = y', is y' really present??
 Definition 2
A function y(x) [or just y] is a PARTICULAR SOLUTION of a D.E. if the D.E. is a true statement about y.
As examples, y = 2x is a particular solution of example [a] above,
y = x3 - 4x -1 is a particular solution of example [b] above,
and any function with a derivative is a particular solution of exampe [d] above.
 Definition 3
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 = x3 - 4x + C is the general solution of example [b] above.
If (as in [b] above), y' = 3x2 - 4 and [y = -4] when [x = 1], then first anti-differentiate [b]
to obtain a GENERAL SOLUTION : y = x3 - 4x + C [as shown in the right figure below].
Then substitute x = 1 and y = -4, to get : -4 = 13 - 4(1) + C. Solve for C to get C = -1.
Therefore, the PARTICULAR SOLUTION for which [y = -4] and [x = 1] is [y = x3 - 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.

The general solution to [b] (named above) is obtained by anti-differentiating both sides of [b]. If you let your mouse wander across the graph below, you will see many of the particular solutions of [b], each having the equation y = x3 - 4x + C, and each labeled with it's own value of "C".

Separable differential equations can
be written in the form f(y)y' = g(x) :
doing this may tax your algebra skills.
An example is at the right :
 Solve the differential equation y' = y2x+1
Solution : first move all y to the left, and all x to the right
 y'y2 = 1x+1
Next : Anti-differentiate both sides
 y'y2 dx = 1x+1 dx
Next : Remove a factor of y' to change variable to y :
 1y2 dy = 1x+1 dx
Finally : antidifferentiate (review methods ?):
 -1 y-1 = (x+1) + C
or
 y = [- (x+1) + C] -1
 Your mouse arrow can control which particular solution of y' = 3x2 - 4, is visible. To keep a chosen solution visible while answering a question below, slide the mouse arrow off either left or right edge of the graph.

 Exercize 4
 Use your mouse on the graph above, to show the particular solution of y' = 3x2 - 4 which satisfies y(0) = 1 (see point A). Then check your choice here:
 Exercize 5
 Use your mouse on the graph above, to show the particular solution of y' = 3x2 - 4 which satisfies y(-1) = 0 (see point B). Then check your choice here:
 Exercize 6
 Use your mouse on the graph above, to show the particular solution of y' = 3x2 - 4 which satisfies y(1) = -1.6 (see point D). Then check your choice here:
 Exercize 7
 Use your mouse on the graph above, to show the particular solution of y' = 3x2 - 4 which satisfies y(-1) = 3.6 (see point E). Then check your choice here: