The solution of the differential equation (d y)/(d x)= y φ^ (x)-y^2...

$$\begin{aligned} & \frac{d y}{d x}=\frac{y \phi^{\prime}(x)-y^2}{\phi(x)}=\frac{y \phi^{\prime}(x)}{\phi(x)}-\frac{y^2}{\phi(x)} \\ & \Rightarrow \frac{d y}{d x}=\frac{y^2 \cdot \phi^{\prime}(x)}{y \cdot \phi(x)}-\frac{y^2}{\phi(x)}\end{aligned}$$$$\begin{aligned} & \Rightarrow \frac{1}{y^2}\left(\frac{d y}{d x}\right)=\frac{\phi^{\prime}(x)}{y \cdot \phi(x)}-\frac{1}{\phi(x)} \\ & \Rightarrow \frac{1}{y^2}\left(\frac{d y}{d x}\right)-\frac{1}{y} \cdot \frac{\phi^{\prime}(x)}{\phi(x)}=\frac{-1}{\phi(x)}\end{aligned}$$$Let$\mathrm{t}=\frac{-1}{\mathrm{y}} \Rightarrow \frac{\mathrm{dt}}{\mathrm{dx}}=\frac{1}{\mathrm{y}^2}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)\Rightarrow \frac{\mathrm{dt}}{\mathrm{dx}}+\mathrm{t} \cdot \frac{\phi^{\prime}(\mathrm{x})}{\phi(\mathrm{x})}=\frac{-1}{\phi(\mathrm{x})}I . F=e^{\int \frac{\phi^{\prime}(x)}{\phi(x)} \cdot d x}=e^{\log \phi(x)}=\phi(x)$Solution of differential equation is$$\begin{aligned} \mathrm{t} \cdot \phi(\mathrm{x}) & =\int \frac{-1}{\phi(\mathrm{x})} \times \phi(\mathrm{x}) \cdot \mathrm{dx} \\ \Rightarrow \frac{-1}{\mathrm{y}} \phi(\mathrm{x}) & =-\mathrm{x}-\mathrm{c} \Rightarrow \frac{\phi(\mathrm{x})}{\mathrm{y}}=\mathrm{x}+\mathrm{c} \Rightarrow \mathrm{y}=\frac{\phi(\mathrm{x})}{\mathrm{x}+\mathrm{c}}\end{aligned}$$