A differential equation containing a homogeneous function is called a homogeneous differential e....

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A differential equation containing a homogeneous function is called a homogeneous differential equation. The function \( f(x, y) \) is called a homogeneous function if \( f(\lambda x, \lambda y)=\lambda^{n} f \)
\( \mathrm{P} \)
\( (x, y) \), for any non-zero constant \( \lambda \). The general form of the homogeneous differential equation is of the form \( f(x, y) \). \( d y+g(x, y) \).

The slope of the tangent at \( (x, y) \) to a curve passing through \( \left(1, \frac{\pi}{4}\right) \) is given by \( \frac{y}{x}-\cos ^{2}\left(\frac{y}{x}\right) \), then the equation of the curve is
(1) \( y=\tan ^{-1}\left[\log \left(\frac{e}{x}\right)\right] \)
(2) \( y=x \tan ^{-1}\left[\log \left(\frac{x}{e}\right)\right] \)
(3) \( y=x \tan ^{-1}\left[\log \left(\frac{e}{x}\right)\right] \)
(4) None of these


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