y-y-y-2-e-y-y-y-2-not-sure-if-it-s-possible-to-solve-this-at-all-

Question Number 46694 by MJS last updated on 30/Oct/18

(y^{″} y - {(y^{'})}^{2}) e^{\frac{y^{'}}{y}} = y^{2}

not sure if {it}^{'} s possible to solve this at all \dots

Answered by ajfour last updated on 30/Oct/18

\ln [y^{″} y - {(y^{'})}^{2}] + \frac{y^{'}}{y} = 2 \ln y

\Rightarrow \ln [\frac{y^{″}}{y} - {(\frac{y^{'}}{y})}^{2}] = - \frac{y^{'}}{y}

l e t \frac{y^{'}}{y} = t \Rightarrow \frac{{(y^{″} y - y^{'})}^{2}}{y^{2}} = \frac{d t}{d x}

\Rightarrow \ln \frac{d t}{d x} = - t

o r d t = e^{- t} d x

\Rightarrow \int e^{t} d t = \int d x

\Rightarrow e^{t} = x + c_{1}

e^{\frac{y^{'}}{y}} = x + c_{1}

\Rightarrow \int \frac{d y}{y} = \int \ln (x + c_{1}) d x

\ln y = x \ln (x + c_{1}) - \int \frac{x d x}{x + c_{1}}

\ln y = x \ln (x + c_{1}) - x + c_{1} \ln (x + c_{1}) + c_{2}

________________________.

Commented by MJS last updated on 30/Oct/18

thank you!

Facebook Tweet Pin