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Category: Differential Equation

x-2-y-xy-y-0-

Question Number 130830 by Ar Brandon last updated on 29/Jan/21 $$\mathrm{x}^{\mathrm{2}} \mathrm{y}''−\mathrm{xy}'+\mathrm{y}=\mathrm{0} \\ $$ Answered by bemath last updated on 29/Jan/21 $$\mathrm{Cauchy}−\mathrm{Euler}\:\mathrm{diff}\:\mathrm{eq} \\ $$$$\mathrm{let}\:\mathrm{y}\:=\:\mathrm{x}^{\mathrm{m}} \:\rightarrow\begin{cases}{\mathrm{y}'=\mathrm{mx}^{\mathrm{m}−\mathrm{1}}…

Question-130301

Question Number 130301 by sarahvalencia last updated on 24/Jan/21 Answered by benjo_mathlover last updated on 24/Jan/21 $$\left(\mathrm{2}\right)\:\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{y}^{\mathrm{2}} }\:\Rightarrow\:\int\:\mathrm{y}^{\mathrm{2}} \mathrm{dy}−\int\mathrm{x}^{\mathrm{2}} \mathrm{dx}=\mathrm{C} \\ $$$$\:\mathrm{y}^{\mathrm{3}} −\mathrm{x}^{\mathrm{3}} \:=\:\mathrm{3C}\:;\:\mathrm{y}^{\mathrm{3}}…

Given-8-x-9-x-dy-dx-4-9-x-x-gt-1-and-y-0-7-Find-y-256-

Question Number 129949 by bemath last updated on 21/Jan/21 $$\mathrm{Given}\:\mathrm{8}\sqrt{\mathrm{x}}\:\left(\sqrt{\mathrm{9}+\sqrt{\mathrm{x}}}\:\right)\mathrm{dy}\:=\:\frac{\mathrm{dx}}{\:\sqrt{\mathrm{4}+\sqrt{\mathrm{9}+\sqrt{\mathrm{x}}}}}\:,\:\mathrm{x}>\mathrm{1} \\ $$$$\mathrm{and}\:\mathrm{y}\left(\mathrm{0}\right)=\sqrt{\mathrm{7}}\:.\:\mathrm{Find}\:\mathrm{y}\left(\mathrm{256}\right). \\ $$ Answered by liberty last updated on 21/Jan/21 $$\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\frac{\mathrm{1}}{\mathrm{8}\sqrt{\mathrm{x}}\:\left(\sqrt{\mathrm{9}+\sqrt{\mathrm{x}}}\right)\left(\sqrt{\mathrm{4}+\sqrt{\mathrm{9}+\sqrt{\mathrm{x}}}}\right)} \\ $$$$\:\mathrm{let}\:\sqrt{\mathrm{4}+\sqrt{\mathrm{9}+\sqrt{\mathrm{x}}}\:}\:=\:\mathrm{z}\:;\:\sqrt{\mathrm{9}+\sqrt{\mathrm{x}}}\:=\:\mathrm{z}^{\mathrm{2}} −\mathrm{4}…