Question Number 210017 by klipto last updated on 28/Jul/24
$$ \\ $$$$\boldsymbol{\mathrm{MATH}}−\boldsymbol{\mathrm{WHIZZKID}} \\ $$$$\boldsymbol{\mathrm{using}}\:\boldsymbol{\mathrm{kamke}}\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{genral}} \\ $$$$\boldsymbol{\mathrm{solution}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{differential}}\:\boldsymbol{\mathrm{equation}} \\ $$$$\mathrm{1}.\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} \boldsymbol{\mathrm{y}}''+\boldsymbol{\mathrm{x}}^{\mathrm{2}} \boldsymbol{\mathrm{y}}'−\mathrm{2}\boldsymbol{\mathrm{y}}=\mathrm{0} \\ $$$$−−−−−−−−− \\ $$$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{this}}\:\boldsymbol{\mathrm{using}}\:\boldsymbol{\mathrm{forbenius}}\:\boldsymbol{\mathrm{mtd}} \\ $$$$\mathrm{1}.\boldsymbol{\mathrm{x}}^{\mathrm{2}} \boldsymbol{\mathrm{y}}''+\left(\boldsymbol{\mathrm{x}}^{\mathrm{3}} −\mathrm{3}\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{y}}'+\left(\mathrm{4}−\mathrm{2}\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{y}}=\mathrm{0} \\ $$$$−−−−−−−− \\ $$$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{differential}}\:\boldsymbol{\mathrm{eqn}}\:\boldsymbol{\mathrm{by}}\:\boldsymbol{\mathrm{power}}\:\boldsymbol{\mathrm{series}} \\ $$$$\mathrm{1}.\:\boldsymbol{\mathrm{y}}''−\mathrm{2}\boldsymbol{\mathrm{xy}}'+\mathrm{2}\boldsymbol{\mathrm{py}}=\mathrm{0} \\ $$$$−−−−−−−−− \\ $$$$\boldsymbol{\mathrm{use}}\:\boldsymbol{\mathrm{perseval}}'\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{theorem}}\:\boldsymbol{\mathrm{to}} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \left(\boldsymbol{\alpha}\frac{\boldsymbol{\pi}}{\mathrm{2}}\right)}{\left(\mathrm{1}−\boldsymbol{\alpha}^{\mathrm{2}} \right)^{\mathrm{2}} }\boldsymbol{\mathrm{dx}}. \\ $$$$−−−−−−−−−− \\ $$$$\boldsymbol{\mathrm{evaluate}}\:\boldsymbol{\mathrm{this}}\:\boldsymbol{\mathrm{integral}}\:\boldsymbol{\mathrm{by}}\:\boldsymbol{\mathrm{contour}}\:\boldsymbol{\mathrm{integration}} \\ $$$$\mathrm{1}.\:\int_{\mathrm{0}} ^{\infty} \frac{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \left(\boldsymbol{\alpha}\frac{\boldsymbol{\pi}}{\mathrm{2}}\right)}{\left(\mathrm{1}−\boldsymbol{\alpha}^{\mathrm{2}} \right)^{\mathrm{2}} }\boldsymbol{\mathrm{dx}}. \\ $$$$−−−−−−−−− \\ $$$$\oint_{\boldsymbol{\mathrm{c}}} \frac{\mathrm{1}+\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{i}\pi\mathrm{z}}} }{\left(\boldsymbol{\mathrm{z}}−\mathrm{1}\right)^{\mathrm{2}} \left(\boldsymbol{\mathrm{z}}+\mathrm{1}\right)^{\mathrm{2}} }\boldsymbol{\mathrm{dz}} \\ $$$$\boldsymbol{\mathrm{c}}−\mathrm{upper}\:\mathrm{half}\:\mathrm{plane} \\ $$$$\boldsymbol{\mathrm{klipto}}−\boldsymbol{\mathrm{quanta}}\biguplus \\ $$