MTH 425
2022-04-21
2.2 Separable DE
Consider
dy
= f ( x, y )
dx
dy
Separable DE
If f does not depend on y ,
= g ( x ), can be solved
dx
by integration
()
()
y = ò g x dx = G x + c
Text: Sec 2.2
where G ( x ) is an anti-derivative (indefinite integral)
Some functions, termed non-elementary, do not possess an
anti-derivative that is an elementary function
1
1
2
2
Separable DE
Example 2.2.1
Solve
A 1st order DE of the form
dy
= g ( x) h ( y)
dx
(1 + x)dx − ydy = 0
a)
dy
= (1 + x) 2
dx
dy
+ 2 xy 2 = 0
dx
dy xy + 2 y − x − 2
=
dx xy − 3 y + x − 3
b)
is said to be separable, or have separable variables
c)
A separable equation can be rewritten in the form
p ( y ) dy = g ( x ) dx
d)
which is solved by integrating both sides
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3
4
4
Example 2.2.1
a)
c)
Example 2.2.2
(1 + x)dx − ydy = 0
Solve the Initial Value Problem
dy
+ 2 xy 2 = 0
dx
5
5
LF_S22
a)
dy y 2 − 1
=
;
dx x 2 − 1
b)
(1 + x 4 )dy + x(1 + 4 y 2 )dx = 0;
d)
sin xdx + ydy = 0;
y(0) = 1
c)
dy
= y 2 sin( x 2 );
dx
y(−2) = 13
y (2) = 2
y(1) = 0
8
8
1
MTH 425
2022-04-21
Example 2.2.2
a)
dy y 2 − 1
=
;
dx x 2 − 1
y (2) = 2
Question?
9
9
LF_S22
11
11
2