Step 1:

Let $P(x,y)$ be any point on the given curve.

The slope of the tangent at P=$\lambda$(slope of line joining the point P and the origin)

(i.e) $\large\frac{dy}{dx}=$$\lambda\big(\large\frac{y-0}{x-0}\big)$

$\Rightarrow \large\frac{dy}{dx}=$$\lambda \large\frac{y}{x}$

This is the required differential equation.

$\large\frac{dy}{dx}=\frac{\lambda y}{x}$

Step 2:

Separating the variables we get,

$\large\frac{dy}{y}=$$\lambda \large\frac{dx}{x}$

Integrating on both sides we get,

$\int \large\frac{dy}{y}=$$\lambda\int\large\frac{dx}{x}$

(i.e)$\log y=\lambda \log x+\log C$

$\Rightarrow \log y=\log x^{\lambda}.C$

$y=Cx^{\lambda}$

This is the equation of the required curve.