(1)
or
x2 + y2 + 2ax – 2ay + a2 = 0 (2)
Differentiating equation (2) with respect to x, we get
2
2
2
2
dy
dy
x
y
a
a
dx
dx
+
+
−
 = 0
or
dy
x
+y dx
 =
1
dy
a dx
⎛
−⎞
⎜
⎟
⎝
⎠
or
a =
1
x
y y
y
′
+
′ −
Substituting the value of a in equation (1), we get
2
2
1
1
x
y y
x
y y
x
y
y
y
′
′
+
+
⎡
⎤
⎡
⎤
+
+
−
⎢
⎥
⎢
⎥
′
′
−
−
⎣
⎦
⎣
⎦
 = 
2
1
x
yy y
′
+
⎡
⎤
⎢
⎥
′ −
⎣
⎦
or
[xy′ – x + x + y y′]2 + [y y′ – y – x – y y′]2 = [x + y y′]2
or
(x + y)2 y′2 + [x + y]2 = [x + y y′]2
or
(x + y)2 [(y′)2 + 1] = [x + y y′]2
which is the differential equation representing the given family of circles Example 26 Find the particular solution of the differential equation log
3
4
dy
x
y
⎛dx
⎞ =
+
⎜
⎟
⎝
⎠
given that y = 0 when x = 0 Solution The given differential equation can be written as
dy
dx  = e(3x + 4y)
or
dy
dx  = e3x