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Question Number 210017 by klipto last updated on 28/Jul/24

$$$$$$\mathbf{MATH}-\mathbf{WHIZZKID}$$$$\mathbf{using}\mathbf{kamke}\mathbf{find}\mathbf{the}\mathbf{genral}$$$$\mathbf{solution}\mathbf{for}\mathbf{the}\mathbf{differential}\mathbf{equation}$$$$1.{\mathbf{x}}^2{\mathbf{y}}^{″}+{\mathbf{x}}^2{\mathbf{y}}^{\prime }-2\mathbf{y}=0$$$$---------$$$$\mathbf{solve}\mathbf{this}\mathbf{using}\mathbf{forbenius}\mathbf{mtd}$$$$1.{\mathbf{x}}^2{\mathbf{y}}^{″}+({\mathbf{x}}^3-3\mathbf{x}){\mathbf{y}}^{\prime }+(4-2\mathbf{x})\mathbf{y}=0$$$$--------$$$$\mathbf{solve}\mathbf{the}\mathbf{differential}\mathbf{eqn}\mathbf{by}\mathbf{power}\mathbf{series}$$$$1.{\mathbf{y}}^{″}-2{\mathbf{xy}}^{\prime }+2\mathbf{py}=0$$$$---------$$$$\mathbf{use}{\mathbf{perseval}}^{\prime }\mathbf{s}\mathbf{theorem}\mathbf{to}$$$${\int }_0^{\mathrm{\infty }}\frac{{\mathbf{cos}}^2\left(\mathit{\alpha }\frac{\mathit{\pi }}{2}\right)}{{(1-{\mathit{\alpha }}^2)}^2}\mathbf{dx}.$$$$----------$$$$\mathbf{evaluate}\mathbf{this}\mathbf{integral}\mathbf{by}\mathbf{contour}\mathbf{integration}$$$$1.{\int }_0^{\mathrm{\infty }}\frac{{\mathbf{cos}}^2\left(\mathit{\alpha }\frac{\mathit{\pi }}{2}\right)}{{(1-{\mathit{\alpha }}^2)}^2}\mathbf{dx}.$$$$---------$$$${\oint }_{\mathbf{c}}\frac{1+{\mathbf{e}}^{\mathbf{i}\mathit{\pi }\mathbf{z}}}{{(\mathbf{z}-1)}^2{(\mathbf{z}+1)}^2}\mathbf{dz}$$$$\mathbf{c}-\mathrm{upper}\mathrm{half}\mathrm{plane}$$$$\mathbf{klipto}-\mathbf{quanta}⨄$$

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