Question

The family of curves which satisfies the differential equation $y{e}^{\frac{x}{y}}dx=(x{e}^{\frac{x}{y}}+y^2)dy,(y\ne 0)$ is
(where ${}^{\prime }{C}^{\prime }$ is the constant of integration )

• A. $y{e}^{\frac{x}{y}}=C$
• B. $y+e^{\frac{x}{y}}=C$
• C. $x-e^{\frac{y}{x}}=C$
• D. $y-e^{\frac{x}{y}}=C$

Answer 1

The correct option is D $y-e^{\frac{x}{y}}=C$

$y{e}^{\frac{x}{y}}dx=(x{e}^{\frac{x}{y}}+y^2)dy,(y\ne 0)$
$\Rightarrow \frac{dx}{dy}=\frac{x{e}^{\frac{x}{y}}+y^2}{y{e}^{\frac{x}{y}}}$
$\text{Put}x=vy$
$\frac{dx}{dy}=v+y\frac{dv}{dy}$
$\therefore v+y\frac{dv}{dy}=v+\frac{y}{e^v}$
$\Rightarrow \int e^{v}dv=\int dy$
$\Rightarrow e^v+C=y$
$\Rightarrow y-e^{\frac{x}{y}}=C$