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Question Number 65044 by mathmax by abdo last updated on 24/Jul/19

$$solve{x}^2{y}^{{}^{″}}+{xy}^{{}^{\prime }}+y=0on]0,+\mathrm{\infty }[(putx=e^t)$$

Commented by mathmax by abdo last updated on 08/Aug/19

$$letx=e^t\Rightarrow y\left(x\right)=y\left(e^t\right)=z\left(t\right)\Rightarrow y\left(x\right)=z\left(lnx\right)\Rightarrow$$$$y^{{}^{\prime }}=\frac{1}{x}{z}^{{}^{\prime }}\left(lnx\right)and{y}^{{}^{″}}=-\frac{1}{x^2}{z}^{{}^{\prime }}\left(lnx\right)+\frac{1}{x^2}{z}^{{}^{″}}\left(lnx\right)$$$$\left(e\right)\Rightarrow x^2(-\frac{1}{x^2}{z}^{{}^{\prime }}\left(lnx\right)+\frac{1}{x^2}{z}^{{}^{″}}\left(lnx\right))+z^{{}^{\prime }}\left(lnx\right)+z\left(lnx\right)=0\Rightarrow$$$$\Rightarrow z^{{}^{″}}\left(t\right)+z\left(t\right)=0\to ce\to r^2+1=0\Rightarrow r=\stackrel{-}{+}i\Rightarrow$$$$z=\alpha e^{it}+\beta e^{-it}\Rightarrow z\left(t\right)=acost+bsint\Rightarrow$$$$y\left(x\right)=acos\left(lnx\right)+bsin\left(lnx\right).$$

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