F =
1 dx
x
e
e
  
 
Multiplying both sides of equation by I F, we get
x
x
dy
e
e
y
dx
−
−−
 = e–x cos x
or
(
dy yex)
dx
−
= e–x cos x
On integrating both sides with respect to x, we get
ye– x =
cos
C
ex
x dx
−
+
∫ (1)
Let
I =
excos
x dx
−
∫
= cos
( sin ) (
)
1
x
x
xe
x
e
dx
−
−
⎛
⎞ −
−
−
⎜
⎝−⎟
⎠ ∫
© NCERT
not to be republished
MATHEMATICS
410
=
cos
sin
x
x
x e
x e
dx
−
−
−
−∫
=
cos
sin (–
)
cos
(
)
x
x
x
x e
x
e
x
e
dx
−
−
−
⎡
⎤
−
−
−
−
⎣
⎦
∫
=
cos
sin
cos
x
x
x
x e
x e
x e
dx
−
−
−
−
+
−∫
or
I = – e–x cos x + sin x e–x – I
or
2I = (sin x – cos x) e–x
or
I = (sin
cos )
2
x
x
x e−
−
Substituting the value of I in equation (1), we get
ye– x =
sin
cos
C
2
x
x
x e−
−
⎛
⎞
+
⎜
⎟
⎝
⎠
or
y =
sin
cos
C
2
x
x
x
e
−
⎛
⎞ +
⎜
⎟
⎝
⎠
which is the general solution of the given differential equation Example 20  Find the general solution of the differential equation 
2
2
(
0)
xdy
y
x
x
dx +
=
≠