Practicing Success
If $\phi(x)$ is a differentiable function, then the solution of the differential equation |
$y=\{\phi(x)-1\}+C e^{-\phi(x)}$ $y \phi(x)=\{\phi(x)\}^2+C$ $y e^{\phi(x)}=\phi(x) e^{\phi(x)}+C$ $y-\phi(x)=\phi(x) e^{-\phi(x)}$ |
$y=\{\phi(x)-1\}+C e^{-\phi(x)}$ |
We have, $d y+\left\{y \phi'(x)-\phi(x) \phi'(x)\right\} d x=0$ $\Rightarrow \frac{d y}{d x}+\phi'(x) . y=\phi(x) \phi'(x)$ .....(i) This is a linear differential equation with Integrating factor = $e^{\int \phi'(x) d x}=e^{\phi(x)}$ Multiplying (i) by $\phi(x)$ and integrating, we get $y e^{\phi(x)} =\int \phi(x) \phi'(x) e^{\phi(x)} d x$ $\Rightarrow y e^{\phi(x)} =\int e^{\phi(x)} \phi(x) \phi'(x) d x$ $\Rightarrow y e^{\phi(x)}=\int \phi(x) e^{\phi(x)} \phi'(x) d x$ $\Rightarrow y e^{\phi(x)}=\phi(x) e^{\phi(x)}-\int \phi'(x) e^{\phi(x)} d x$ $\Rightarrow y e^{\phi(x)}=\phi(x) e^{\phi(x)}-e^{\phi(x)}+C$ $\Rightarrow y=(\phi(x)-1)+C e^{-\phi(x)}$ |