Page 145 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes

Task Solve the following differential equation:

dy y  1
x
 .
x
dx y  5
Self Assessment

Fill in the blanks:
12. A differential equation of the form ........................... can be reduced to the homogeneous
form.
dy ax  by c

13. A differential equation of the form  is not homogeneous, but we can

dx Ax  By C
formulate it so by a transformation in the ........................... of x and y.

dy ax  by c
14. When ..........................., then the equation of the form  becomes
dx Ax  By C

dY aX  bY  ah bk c 


 .

dX AX  BY  Ah Bk c 


dy ax  by c
15. When ..........................., then the equation of the form  becomes
dx Ax  By C


dy ax by   c

   f ax by .

dx m ax by   c

10.3 Summary
 Homogeneous equation is just an equation where both coefficients of the differentials dx
and dy are homogeneous.
 Homogeneous functions redefined as functions where the sums of the powers of each
term are the same.
 A homogeneous equation can be malformed into a distinguishable equation by a change
of variables.

dy f 1  ,x y 
 An equation of the form  is called a homogeneous function of the same
dx f 2  ,x y 
degree in x and y.

 If you identify the truth that an equation is homogeneous you can, in some cases, carry out
a substitution which will permit you to apply separation of variables to solve the equation.

 If f (x, y) and f (x, y) are homogeneous functions of degree n in x and y, then
1 2
y
y
f 1  ,x y   x n  1   and f 2  ,x y  x n  2  
 .
 
x
x
   
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