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Find the particular solution, satisfying the given condition, for the following differential equation: `(dy)/(dx)-y/x+cos e c\ (y/x)=0; y=0` when `x=1`

Text Solution

Verified by Experts

The given differential equation may be written as
`(dy)/(dx)=y/x=-"cosec"y/x`………….(i)
This is the form `(dy)/(dx)=f(y/x)`. So, it is homogeneous.
Putting `y=vx` and `(dy)/(dx)=v+x(dv)/(dx)` in (i), we get
`v+x(dv)/(dx)=v-"cosec"v`
`rArr -sinvdv=1/xdx`
`rArr cosv=log|x|+C`, where C is an arbitary constant.
`rArr cosy/x=log|x|+C...................(ii), [therefore v=y/x]`.
Putting `x=1` and y=0 in (ii), we get C=1.
Hence, `cosy/x=1+log|x|` is the required solution.
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