(1)
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MATHEMATICS
402
It is a differential equation of the form 
F( ,
)
dy
x y
dx = Here
F(x, y) =
cos
cos
y
y
x
x
y
x
x
⎛
⎞ +
⎜
⎟
⎝
⎛⎠
⎞
⎜
⎟
⎝
⎠
Replacing x by λx and y by λy, we get
F(λx, λy) =
0
[ cos
]
[F( , )]
cos
y
y
x
x
x y
y
x
x
⎛
⎞
λ
+
⎜
⎟
⎝
⎠
= λ
⎛
⎞
λ⎜
⎟
⎝
⎠
Thus, F(x, y) is a homogeneous function of degree zero Therefore, the given differential equation is a homogeneous differential equation To solve it we make the substitution
y = vx