Question Number 219996 by SdC355 last updated on 04/May/25 $$\mathrm{Solve}\:\mathrm{Equation} \\ $$$$\frac{\mathrm{d}{x}\left({t}\right)}{\mathrm{d}{t}}=\mathrm{2}{x}\left({t}\right)+{y}\left({t}\right) \\ $$$$\frac{\mathrm{d}{y}\left({t}\right)}{\mathrm{d}{t}}=−\mathrm{3}{y}\left({t}\right) \\ $$$$\begin{pmatrix}{{x}^{\left(\mathrm{1}\right)} \left({t}\right)}\\{{y}^{\left(\mathrm{1}\right)} \left({t}\right)}\end{pmatrix}=\begin{pmatrix}{\mathrm{2}}&{\:\:\:\:\mathrm{1}}\\{\mathrm{0}}&{−\mathrm{3}}\end{pmatrix}\begin{pmatrix}{{x}\left({t}\right)}\\{{y}\left({t}\right)}\end{pmatrix} \\ $$$$\mathrm{A}=\begin{pmatrix}{\mathrm{2}}&{\:\:\:\:\mathrm{1}}\\{\mathrm{0}}&{−\mathrm{3}}\end{pmatrix} \\ $$$$\mathrm{det}\left\{\mathrm{A}−\boldsymbol{\lambda}\mathrm{E}\right\}=\mathrm{0} \\ $$$$\mathrm{det}\left\{\begin{pmatrix}{\mathrm{2}}&{\:\:\:\:\mathrm{1}}\\{\mathrm{0}}&{−\mathrm{3}}\end{pmatrix}−\begin{pmatrix}{\boldsymbol{\lambda}}&{\mathrm{0}}\\{\mathrm{0}}&{\boldsymbol{\lambda}}\end{pmatrix}\right\}=\mathrm{0} \\…
Question Number 219998 by fantastic last updated on 04/May/25 Answered by mehdee7396 last updated on 04/May/25 $${AB}=\sqrt{\mathrm{61}}\:\:\&\:\:\:{AC}=\mathrm{6}\sqrt{\mathrm{10}} \\ $$$${S}_{{ABC}} =\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{13}×\mathrm{6}=\mathrm{39} \\ $$$${R}=\frac{{abc}}{\mathrm{4}{S}}=\frac{\mathrm{13}×\sqrt{\mathrm{61}}×\mathrm{6}\sqrt{\mathrm{10}}}{\mathrm{4}×\mathrm{39}}=\frac{\sqrt{\mathrm{610}}}{\mathrm{2}} \\ $$$$ \\…
Question Number 219988 by Nicholas666 last updated on 04/May/25 $$ \\ $$$$\:\:\:\:\mathrm{let}\:{s}>\mathrm{1}\:\mathrm{be}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number}.\:\mathrm{for}\:\mathrm{all}\:\mathrm{continues}\:\mathrm{function}\:{f}:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\mathbb{R} \\ $$$$\:\:\:\mathrm{such}\:\mathrm{that}\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} {f}\left({x}\right)=\mathrm{0},\:\mathrm{determind}\:\mathrm{of}\:\mathrm{the}\:\mathrm{exist}\:\mathrm{a} \\ $$$$\:\:\:\:\:\mathrm{positive}\:\mathrm{constant}\:{K}\left({s}\right)\:\mathrm{statisfying}: \\ $$$$\:\:\:\left(\int_{\mathrm{0}} ^{\:\mathrm{1}} {f}\left({x}\right)\centerdot\mathrm{Li}_{{s}} \left({x}\right){dx}\right)^{\mathrm{2}} \geqslant{K}\left({s}\right)\int_{\:\mathrm{0}} ^{\:\mathrm{1}}…
Question Number 220040 by Nicholas666 last updated on 04/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\mathrm{0}} {\int}^{\:\infty} \:\frac{\mathrm{1}}{\:\sqrt{{x}^{\mathrm{8}} \:+\:\mathrm{1}}}\:{dx} \\ $$$$ \\ $$ Answered by MrGaster last updated on…
Question Number 220037 by Nicholas666 last updated on 04/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\frac{\mathrm{1}}{\:\sqrt{{x}^{\mathrm{8}} \:+\:\mathrm{1}}}\:{dx} \\ $$$$ \\ $$ Answered by MrGaster last updated on 04/May/25 $$\int\:\frac{\mathrm{1}}{\:\sqrt{{x}^{\mathrm{8}}…
Question Number 220038 by Nicholas666 last updated on 04/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \frac{\mathrm{1}}{\:\sqrt{{x}^{\mathrm{8}} \:+\:\mathrm{1}}}\:{dx} \\ $$$$\: \\ $$ Answered by MrGaster last updated on…
Question Number 220034 by Tawa11 last updated on 04/May/25 Answered by efronzo1 last updated on 04/May/25 $$\:\left(\mathrm{1}\right)\:\mathrm{x}^{\mathrm{2}} \:=\:\mathrm{96}{a} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{96}{b} \\ $$$$\:\left(\mathrm{3}\right)\:{a}+{b}\:=\:\mathrm{96}\Rightarrow{b}=\:\mathrm{96}−{a} \\ $$$$\:\Rightarrow\:\mathrm{36}^{\mathrm{2}}…
Question Number 219970 by Nicholas666 last updated on 04/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:{e}^{{x}^{\mathrm{2}} } \:{dx} \\ $$$$ \\ $$ Answered by MATHEMATICSAM last updated on 04/May/25…
Question Number 219898 by MrGaster last updated on 03/May/25 Answered by Frix last updated on 03/May/25 $$\mathrm{Obviously}\:{x}=\mathrm{8} \\ $$$$\mathrm{No}\:\mathrm{other}\:\mathrm{solution}. \\ $$ Commented by MrGaster last…
Question Number 219899 by MrGaster last updated on 03/May/25 Answered by Frix last updated on 03/May/25 $${y}=\frac{\mathrm{10}−{x}^{\mathrm{3}} }{\mathrm{9}{x}^{\mathrm{2}} } \\ $$$$\Rightarrow \\ $$$${x}^{\mathrm{9}} −\mathrm{201}{x}^{\mathrm{6}} +\mathrm{25}{x}^{\mathrm{3}}…