Question

Find the equation of a curve whose tangent at any point on it, different from origin,has slope y+y/x

Answer 1

$\text{ I have applied variable separable method to find the equation of the curve}$

$\textbf{Given:}$

$\text{Slope of tangent at any point on the curve }=y+\frac{y}{x}$

$\implies\frac{dy}{dx}=y+\frac{y}{x}$

$\frac{dy}{dx}=y(1+\frac{1}{x})$

$\frac{dy}{y}=(1+\frac{1}{x})\;dx$

$\text{Integrating on both sides}$

$\int\frac{dy}{y}=\int(1+\frac{1}{x})\;dx$

$\int\frac{dy}{y}=\int{1}\;dx+\int\frac{1}{x})\;dx$

$logy=x+logx+c$

$logy-logx=x+c$

$log\frac{y}{x}=x+c$

$\frac{y}{x}=e^{x+c}$

$\frac{y}{x}=ke^{x}$

$\implies\boxed{\bf\;y=k\;xe^{x}}$

$\text{which is the equation of the required curve}$

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