Solution Of First Order Equations Worksheets With Answers - S. Ghorai Page 4
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5S. Ghorai
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4.1
How to find intgrating factor
Theorem 2. If (2) is such that
1
∂M
∂N
N
∂y
∂x
is a function of x alone, say F (x), then
µ = e
is a function of x only and is an integrating factor for (2).
2
Example 6. (xy
1)dx + (x
xy)dy = 0
2
Solution: Here M = xy
1 and N = x
xy. Also,
1
∂M
∂N
1
=
N
∂y
∂x
x
Hence, 1/x is an integrting factor. Multiplying by 1/x we find
2
(xy
1)dx + (x
xy)dy
2
= 0
xy
ln x
y
/2 = C
x
Theorem 3. If (2) is such that
1
∂M
∂N
M
∂y
∂x
is a function of y alone, say G(y), then
µ = e
is a function of y only and is an integrating factor for (2).
3
2
Example 7. y
dx + (xy
1)dy = 0
3
2
Solution: Here M = y
and N = xy
1. Also,
1
∂M
∂N
2
=
M
∂y
∂x
y
2
2
Hence, 1/y
is an integrting factor. Multiplying by 1/y
we find
3
2
y
dx + (xy
1)dy
1
= 0
xy +
= C
2
y
y
Comment: Sometimes it may be possible to find integrating factor by inspection. For
this, some known differential formulas are useful. Few of these are given below:
x
ydx
xdy
d
=
2
y
y
y
xdy
ydx
d
=
2
x
x
d(xy) = xdy + ydx
x
ydx
xdy
d ln
=
y
xy
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