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Unit 9: Solution of Differential Equation

curve. The equation of each such curve is a particular solution of the differential equation (1). The Notes
equation of the system of all such curves is the general solution of (1).

9.4.2 Equations in Which the Variables are Separable

A differential equation of first order and first degree is of the form

dy
x
y
 f ( , )  0 ...(1)
dx
which is sometimes written as
Mdx + Ndy = 0, ...(2)
where M and N are functions of x and y or constants and f is some known function of x and y. The
following are some standard methods to solve some first order, first degree equations, which
can be classified in one of the categories given below:
If a first order and first degree equation (2) can be put in the form
f (x) dx + f (y)dy = 0,
1 2
then it is said to be in variable separable form. The solution is obtained by integration, i.e.,

y
x
f 1  ( )dx  f 2  ( )dy  , C
where C is an arbitrary constant.

Notes Many differential equations can be reduced to variable separable form by making
dy


)
suitable substitution. For instance, the equation of the form  ( f ax by c can be
dx
reduced to the form of variable separable by putting ax + by + c = u.
2
2
2
2
Example: Solve (x – yx ) dy + (y +xy ) dx = 0
Solution:
Here (x – yx ) dx + (y + xy ) dx = 0
2
2
2
2
1  y 1  x
or 2 dy  2 dx  0
y x
 1 1   1 1 
or  2   dy   2   dx  0
 y y   x x 
Integrating, we get
1 1
  logy   logx c

y x
   y x 
x
or log       c
y
   xy 
which is the required solution.

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