Mathematics - Differential Equation Question with Solution | TestHub

We have $\frac{dy}{dx}=\frac{f^{\prime }(x)}{f(x)}y-\frac{y^2}{f(x)}$ $\Rightarrow \frac{dy}{dx}-\frac{f^{\prime }(x)}{f(x)}y=-\frac{y^2}{f(x)}$ Divide by $y^2:y^{-2}\frac{dy}{dx}-y^{-1}\frac{f^{\prime }(x)}{f(x)}=-\frac{1}{f(x)}$ Put $y^{-1}=z\Rightarrow -y^{-2}\frac{dy}{dx}=\frac{dz}{dx}$ $-\frac{dz}{dx}-\frac{f^{\prime }(x)}{f(x)}(z)=-\frac{1}{f(x)}$ $\Rightarrow \frac{dz}{dx}+\frac{f^{\prime }(x)}{f(x)}z=\frac{1}{f(x)}$ $IF=e^{\int \frac{f^{\prime }(x)}{f(x)}dx}=e^{\log f(x)}=f(x)$ $\therefore$ The solution is $z(f(x))=\int \frac{1}{f(x)}(f(x))dx+c$ $\Rightarrow y^{-1}(f(x))=x+c\Rightarrow f(x)=y(x+c)$