The Integrating Factor of the differential equation 
22
xdy
y
x
dx −
=
 is
(A) e–x
(B) e–y
(C) 1
x
(D) x
19 The Integrating Factor of the differential equation
2
(1
y) dx
yx
dy
−
+
 = 
( 1
1)
 
 
ay
y
 is
(A)
12
y  1
(B)
2
1
y −1
(C)
112
−y
(D)
2
1
1
y
−
Miscellaneous Examples
Example 24 Verify that the function y = c1 eax cos bx + c2 eax sin bx, where c1, c2 are
arbitrary constants is a solution of the differential equation
(
)
2
2
2
2
2
0
d y
ady
a
b
y
dx
dx
−
+
+
=
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not to be republished
DIFFERENTIAL EQUATIONS
415
Solution The given function is
y = eax [c1 cosbx + c2 sinbx] (1)
Differentiating both sides of equation (1) with respect to x, we get
dy
dx  =
  
 
1
2
1
2
–
sin
cos
cos
sin
ax
ax
e
bc
bx
bc
bx
c
bx
c
bx e
a
 
 
 
or
dy
dx  =
2
1
2
1
[(
)cos
(
)sin
]
eax
bc
ac
bx
ac
bc
bx
+
+
− (2)
Differentiating both sides of equation (2) with respect to x, we get
2
d y2
dx
 =
2
1
2
1
[(
) (
sin
)
(
) ( cos
)]
eax
bc
ac
b
bx
ac
bc
b
bx
 
 
 
 
+ 
2
1
2
1
[(
) cos
(
) sin
]
ax