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
\( \mathrm{P} \)
\( f(x, y) \) is called a homogeneous function if \( f(\lambda x, \lambda y)=\lambda^{n} f \) \( (x, y) \), 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 solution of the differential equation \( \frac{d y}{d x}=\frac{y}{x}+\frac{\phi\left(\frac{y}{x}\right)}{\phi^{\prime}\left(\frac{y}{x}\right)} \) is
(1) \( \phi\left(\frac{y}{x}\right)=k x \)
(2) \( \quad x \phi\left(\frac{y}{x}\right)=k \)
(3) \( \phi\left(\frac{y}{x}\right)=k y \)
(4) \( y \phi\left(\frac{y}{x}\right)=k \)


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