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The slope of the tangent to a curve at a...

The slope of the tangent to a curve at any point `(x ,y)` on its given by `y/x-coty/xdotcosy/x ,(x >0,y >0)` and the curve passes though the point `(1,pi//4)dot` Find the equation of the curve.

Text Solution

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The correct Answer is:
`secy/x+log|x|=sqrt(2)`
`(dy)/(dx)={y/x-cot y/x cosy/x}=f(y/x)`, whch is homogenous.
Putting `y=vx` and `(dy)/(dx)=v+x(dv)/(dx)`, we get
`int sec v tan v dv =-int(dx)/(x) rArr sec v=-log|x|+C`
`therefore sec(y/x)+log|x|=C`.
Putting `x=1` and `y=pi/4`, we get `C=sqrt(2)`.
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