Page 130 - DMTH202_BASIC_MATHEMATICS_II

Unit 9: Solution of Differential Equation

Notes
  1 
y

y

Example: Solve   1   cosy dx  {x  logx x sin } dy  0
  x 
Solution:
This given equation is of the form Mdx + Ndy =0
 1
where M = y   1 x   cosy , N = x + log x – sin y
 M 1  N 1
  1  sin ,  1  sin y
y
y  x x  x
 N  M
  , therefore, the given equation is exact.
x  y 
The solution is given by

 Mdx   (terms of N independent of )dy  C
x

y constant
 1 

or,    1+   cosy dx   o.dy  C
x
y constant  
or, y(x + log x) + x cos y = C.
is the required general solution.

x  x  x
Task Solve the differential equation: (1 e y ) dx   1   e y dy  0.

 y 
Self Assessment

Fill in the blanks:

dy
y
x
0
11. A differential equation of first order and first degree is of the form  f ( , )  is
dx
sometimes written as .........................
12. If a first order and first degree equation can be put in the form f1(x) dx + f2(y)dy = 0, then it
is said to be in ......................... form.
13. Many differential equations can be reduced to variable separable form by making suitable
..........................
14. A first order first degree equation of the form M(x, y)dx + N(x, y) dy = 0 is said to be
......................... if its left hand side quantity is the exact differential of some function u(x, y).
15. The ......................... of the exact equation is given by u(x, y) = C.

9.5 Summary

 A differential equation involves independent variables, dependent variables, their
derivatives and constants.
 The order of a differential equation is the order of highest derivative appearing in the
equation.

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