Page 158 - DMTH202_BASIC_MATHEMATICS_II

Unit 11: Linear Differential Equations of First Order

Notes
1
 dx 1
 I . .  e x  e  log x  .
F
x
 Solution of (2) is
1 1
. z   1 . dx  c
x x
z
or   log x  c
x

1
x
or  c  log ,
xy
which is the required solution.

Example:

dy
3
Solve x  y  xy . …..(1)
dx
Solution:
3
Dividing throughout by y , we have

xy  3 dy  y  2  x
dx

or y  3 dy  1 y  2  1. …..(2)
dx x

Putting y = z
-2
  2y  3 dy  dz
dx dx

or y  3 dy   1 dz .
dx 2 dx
 Equation (2) becomes

1 dz 1
  z 1
2 dx x

dz 2
or  z  2. …..(3)
dx x
which is linear in z.

2
 dx  2log x 1
 . .I F  e x  e  2 .
x

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