Page 141 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes Now we consider the following cases to reduce the given differential equation to the
homogeneous form.

Case I

a b
When  .
A B
Putting x = X + h, y = Y + k; (h & k are constants)
so that dx = dX, dy = dY.
Equation (1) becomes:

dY  a X h   b  Y k c



dX A X h   B  Y k c

dY aX  bY  ah bk c 

or  …..(2)
dX AX  BY  Ah Bk c 


Choosing h & k such that (2) becomes homogeneous.
Thus ah + bk + c = 0 and Ah + Bk + c = 0
h k 1
so that  
bC  Bc cA Ca aB  Ab
bC  Bc cA Ca
 h  ,k 
aB  Ab aB  Ab

  a  b i . .,aB bA  0 
e
 
 A B 
 Equation (2) becomes
dY aX  bY

dX AX  BY
which is homogeneous in X & Y and can be solved by putting Y = vX.

Case II

a b
When  , i.e., aB – bA = 0.
A B
The case (1) fails.
a b 1
Now   (say)
A B M
so that A = ma, B = mb.


dy ax by   c

   f ax by 

dx m ax by   c
 Equation (1) reduces to
which can be solved by putting ax + by = t.

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