Solution of the differential equation $y=x \cdot \frac{dy}{dx}+\frac{b}{\frac{dy}{dx}}$ Solve the following differential equation:
$$y=x \cdot \frac{dy}{dx}+\frac{b}{\frac{dy}{dx}}$$ where $b$ is a real constant.
I tried solving it by taking $\frac{dy}{dx}=t$, then solving quadratic equation for roots. But this method gives complicated differential equation once I write expression for roots. Could someone suggest a better approach.

This is a Clairaut equation.
Therefore, you can differentiate both sides w.r.t $x$ to obtain:
$$\frac{dy}{dx}=x\frac{d^2 y}{dx^2}+\frac{dy}{dx}-\frac{b\cdot \frac{d^2 y}{dx^2}}{\left(\frac{dy}{dx}\right)^2}$$
Factoring gives:
$$\frac{d^2 y}{dx^2}\left(x-\frac{b}{(\frac{dy}{dx})^2}\right)=0$$
Now, solve for the following separately:
$$\frac{d^2 y}{dx^2}=0$$
And:
$$x-\frac{b}{(\frac{dy}{dx})^2}=0$$
