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A DIFFERENTIAL EQUATION (DE) is an equation which "contains" derivatives of some function. The word "contains" is not clearly defined : does example [c] above "contain" y' ?? |
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A function y(x) [ = y] is a PARTICULAR SOLUTION of a DE if the DE is a true statement about y. |
| As examples, y = 4x -3 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 example [c] above. |
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The GENERAL SOLUTION of a D.E. is the set of all of it's particular solutions, often expressed using a constant C (or K) which could have any fixed value. |
| As examples, y = 4x -3 + C is the general solution of example [a] above, and y = x3 - 4x + C is the general solution of example [b] above. |
Many differential equations may be solved by separating the variables x and y
on opposite sides of the equation, then anti-differentiating both sides with respect to x.
When anti-differentiating the side containing y, the facts in the table below may be
useful. Each row of this table uses the chain rule or its alternate
statement integration by substitution (both with u = y)
y
Differentiate
Anti-Differentiate
y '
y
Differentiate
Anti-Differentiate
y
Name
Using language of Derivatives
Using language of Anti-Derivatives
2
d
dx
y 2 = 2y
y'
1
2
y 2 =
y
y' dx3
d
dx
y 3 = 3y 2
y'
1
3
y 3 =
y 2
y' dx-1
d
dx
y -1 = -1y -2
y'
-y -1 =
y -2
y' dx-2
d
dx
y -2 = -2 y -3
y'
1
2
y -2 =
y -3
y' dx½
d
dx
y ½ = ½ y -½
y'
2 y ½ =
y -½
y' dxexp
d
dx
ey = ey
y'
ey =
ey
y' dx
d
dx
y =
1
y y'
y =
1
y
y' dx
Example 1
Solve : xy' = y 2
Solution : y -2
y' =
x -1
y -2
y'
dx =
x -1 dx
- y -1
=
x + C
Example 2
Solve : y' - e x = ye x
Solution : y' = ye x + e x
or y' = e x (y + 1)
or
1
y + 1
y' = e x
1
y + 1
y' dx
=
e x dx
(y + 1) = e x + C
