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Question

(C) The given differential equation is $\left(1+e^{\frac{x}{y}}\right) dx + e^{\frac{x}{y}}\left(1-\frac{x}{y}\right) dy = 0$. Rearranging the terms,w

Vedclass · Accepted Answer

$x=vy$

Vedclass · Answer

(C) The given differential equation is $\left(1+e^{\frac{x}{y}}\right) dx + e^{\frac{x}{y}}\left(1-\frac{x}{y}\right) dy = 0$. Rearranging the terms,we get $\frac{dx}{dy} = -\frac{e^{\frac{x}{y}}\left(1-\frac{x}{y}\right)}{1+e^{\frac{x}{y}}}$. Since the equation involves the term $\frac{x}{y}$,it is a homogeneous differential equation of the form $\frac{dx}{dy} = F\left(\frac{x}{y}\right)$. To solve such an equation,we substitute $x = vy$,where $v$ is a function of $y$. Therefore,the correct substitution is $x = vy$.

The homogeneous differential equation of the form $\left(1+e^{\frac{x}{y}}\right) dx + e^{\frac{x}{y}}\left(1-\frac{x}{y}\right) dy = 0$ can be solved by making the substitution:

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