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Question Number 92398 by jagoll last updated on 06/May/20

$${\mathrm{x}}^3\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)+{\mathrm{x}}^2{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2=\ln \left(\mathrm{x}\right)$$

Answered by Joel578 last updated on 06/May/20

Commented by Joel578 last updated on 06/May/20

$$\mathrm{I}\mathrm{can}\mathrm{only}\mathrm{find}\mathrm{the}\mathrm{homogeneous}\mathrm{solution}$$$${\phi }^{″}+x^{-1}\phi =0\to GeneralizedBesselequation$$$$\Rightarrow \phi \left(x\right)=c_1\sqrt{x}{J}_1\left(2\sqrt{x}\right)+c_2\sqrt{x}{Y}_1\left(2\sqrt{x}\right)$$$$\mathrm{with}{J}_1\mathrm{and}{Y}_1\mathrm{are}\mathrm{Bessel}\mathrm{and}\mathrm{Neumann}\mathrm{function}\mathrm{order}1$$

Commented by Joel578 last updated on 06/May/20

$$\mathrm{Maybe}\mathrm{someone}\mathrm{knows}\mathrm{a}\mathrm{better}\mathrm{method}\mathrm{to}$$$$\mathrm{solve}\mathrm{this}\mathrm{diff}\mathrm{equation}$$

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