Mathematics - Differential Equation Question with Solution | TestHub

Given differential equation is $\mathrm{dy}+\left\{y{\varphi }^{\prime }(x)-\varphi (x){\varphi }^{\prime }(x)\right\}dx=0$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}+{\varphi }^{\prime }(\mathrm{x})\mathrm{y}=\varphi (\mathrm{x}){\varphi }^{\prime }(\mathrm{x})$ which is a linear differential equation with $P={\varphi }^{\prime }(x),Q=\varphi (x)\cdot {\varphi }^{\prime }(x)$ and $I\cdot F=e^{\int {\varphi }^{\prime }(x)dx}=e^{\varphi (x)}$ $\therefore$ Solution is $y\cdot e^{\varphi (x)}=\int \varphi (x)\cdot {\varphi }^{\prime }(x)e^{\varphi (x)}dx+C$ $\Rightarrow y\cdot e^{\varphi (x)}=\int \varphi (x)\cdot e^{\varphi (x)}{\varphi }^{\prime }(x)dx+C$ $\Rightarrow y\cdot e^{\varphi (x)}=\varphi (x)e^{\varphi (x)}-\int {\varphi }^{\prime }(x)e^{\varphi (x)}dx+C$ $\Rightarrow y\cdot e^{\varphi (x)}=\varphi (x)e^{\varphi (x)}-e^{\varphi (x)}+C$ $\Rightarrow y=\lceil \varphi (x)-1]+C{e}^{-\varphi (x)}$