Page 139 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes Solution:
x  
e y  1  x 
dx  y 
Here   ……….(1)
dy x y
1  e
dx dv
v
Putting x = vy, so that   y
xy xy
 (1) reduces to

dv e v  1 v 
v  y   v
dy 1  e

dv e v  1 v  v  e v
v
or v   v    v
dy 1  e 1  e
dy 1  e v
or  v dv
y v  e

Integrating, we get
v
log c – log y = log (v + e )
or log y + log (v + e ) = log c
v
or y(v+e) = c
or x + y e = c.
e/y
which is the required general solution.

!
y
Caution When a differential equation contain a number of times, solve it like a
x
y
homogeneous equations by putting  v .
x

Notes The function f does not rely on x & y individually but only on their proportion
y/x or x/y.

Task Solve the following differential equation:

dy dy
y  x   y .
x
dx dx

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