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Solve-the-Differential-equation-x-dy-dx-y-2y-lnx-lny-




Question Number 181218 by Mastermind last updated on 23/Nov/22
Solve the Differential equation:  x(dy/dx)−y=2y(lnx−lny)    .
$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{Differential}\:\mathrm{equation}: \\ $$$$\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}−\mathrm{y}=\mathrm{2y}\left(\mathrm{lnx}−\mathrm{lny}\right) \\ $$$$ \\ $$$$. \\ $$
Answered by qaz last updated on 23/Nov/22
y=zx  x(dy/dx)−y=x((xdz+zdx)/dx)−zx=−2zxlnz  ⇒(dz/(zlnz))=−(2/x)dx  ⇒lnz=Cx^(−2)   ⇒z=e^(Cx^(−2) )   ⇒y=xe^(Cx^(−2) )
$${y}={zx} \\ $$$${x}\frac{{dy}}{{dx}}−{y}={x}\frac{{xdz}+{zdx}}{{dx}}−{zx}=−\mathrm{2}{zxlnz} \\ $$$$\Rightarrow\frac{{dz}}{{zlnz}}=−\frac{\mathrm{2}}{{x}}{dx} \\ $$$$\Rightarrow{lnz}={Cx}^{−\mathrm{2}} \\ $$$$\Rightarrow{z}={e}^{{Cx}^{−\mathrm{2}} } \\ $$$$\Rightarrow{y}={xe}^{{Cx}^{−\mathrm{2}} } \\ $$
Commented by universe last updated on 23/Nov/22
good solution sir
$${good}\:{solution}\:{sir} \\ $$

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