Question Number 63152 by Rio Michael last updated on 29/Jun/19 $${The}\:{Variables}\:{x}\:{and}\:{y}\:{satisfy}\:{the}\:{differential}\:{equation}\: \\ $$$$\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }−{x}\frac{{dy}}{{dx}}\:+\:{y}\:=\:{x}^{\mathrm{2}} \:\:{use}\:{the}\:{approximations} \\ $$$$\:\:\:\left(\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\right)_{{n}} \approx\:\frac{{y}_{{n}+\mathrm{1}} −\mathrm{2}{y}_{{n}} +{y}_{{n}−\mathrm{1}} }{{h}^{\mathrm{2}\:} }\:{and}\:\:\left(\frac{{dy}}{{dx}}\right)\approx\frac{{y}_{{n}+\mathrm{1}}…
Question Number 128684 by Study last updated on 09/Jan/21 $${how}\:{we}\:{can}\:{convert}\:\mathrm{0}.\overset{−} {\mathrm{9}}\:{to}\:\frac{{q}}{{p}}? \\ $$ Answered by liberty last updated on 09/Jan/21 $$\:\mathrm{let}\:\mathrm{r}\:=\:\mathrm{0},\mathrm{99999999999}… \\ $$$$\:\:\:\:\:\:\:\mathrm{10r}\:=\:\mathrm{9}.\mathrm{9999999999}… \\ $$$$\mathrm{substract}\:\Rightarrow\:\mathrm{9r}\:=\:\mathrm{9}\:\Rightarrow\:\mathrm{r}\:=\:\mathrm{1}\:=\:\frac{\mathrm{1}}{\mathrm{1}}\:=\:\frac{\mathrm{2}}{\mathrm{2}}=\frac{\mathrm{n}}{\mathrm{n}}\:;\:\mathrm{n}\neq\:\mathrm{0}…
Question Number 128680 by mnjuly1970 last updated on 09/Jan/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{nice}\:\:{calculus}… \\ $$$$\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{3}{n}+\mathrm{1}\right)\varphi^{\mathrm{3}{n}+\mathrm{1}} }\:=? \\ $$$$\varphi\:::\:\:{golden}\:\:{ratio}… \\ $$$$ \\ $$ Answered by mr W…
Question Number 128681 by bemath last updated on 09/Jan/21 $$\:\mathrm{Given}\:\sqrt[{\mathrm{3}}]{\mathrm{sin}\:\mathrm{x}}\:−\:\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\mathrm{x}}\:=\:\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{\mathrm{3}}} \\ $$$$\:\mathrm{then}\:\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}\:=?\: \\ $$ Commented by MJS_new last updated on 09/Jan/21 $$\mathrm{not}\:\mathrm{funny}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{exactly}\:\mathrm{but}\:\mathrm{possible}… \\ $$…
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Question Number 128674 by bemath last updated on 09/Jan/21 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{3tan}\:\mathrm{4x}−\mathrm{12tan}\:\mathrm{x}}{\mathrm{3sin}\:\mathrm{4x}−\mathrm{12sin}\:\mathrm{x}}\:=\:? \\ $$$$ \\ $$ Answered by liberty last updated on 09/Jan/21 $$\:\mathrm{Taylor}\:\mathrm{series}\:\begin{cases}{\mathrm{tan}\:\mathrm{x}=\mathrm{x}+\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{3}}+\frac{\mathrm{2x}^{\mathrm{5}} }{\mathrm{15}}+…}\\{\mathrm{tan}\:\mathrm{4x}=\mathrm{4x}+\frac{\mathrm{64x}^{\mathrm{3}}…
Question Number 63139 by Tawa1 last updated on 29/Jun/19 $$\underset{\mathrm{n}\:=\:\mathrm{1}} {\overset{\mathrm{m}} {\sum}}\:\frac{\mathrm{log}\:\mathrm{n}}{\mathrm{n}^{\mathrm{3}/\mathrm{2}} } \\ $$ Commented by Tawa1 last updated on 30/Jun/19 $$\mathrm{please}\:\mathrm{help}\:\mathrm{with}\:\mathrm{this}\:\mathrm{sir} \\ $$$$\:\mathrm{Test}\:\mathrm{for}\:\mathrm{convergence}.\:\:\:\:\:\underset{\mathrm{n}\:=\:\mathrm{1}}…
Question Number 63137 by Rio Michael last updated on 29/Jun/19 $${please}\:{who}\:{can}\:{prove}\:{the}\:{half}−{life}\:{in}\:{radioactivity}\:{formula} \\ $$$${of}\:\:{t}_{\frac{\mathrm{1}}{\mathrm{2}}} =\:\frac{{ln}\mathrm{2}}{\lambda}\:\:{where}\:\lambda\:{is}\:{distergration}\:{rate}.\:{Involving}\: \\ $$$${ln}\frac{{N}_{\mathrm{0}} }{{N}_{{t}} }=\:\lambda{t} \\ $$ Answered by Hope last updated…
Question Number 128667 by sarahvalencia last updated on 09/Jan/21 Commented by liberty last updated on 09/Jan/21 $$\:\left(\mathrm{1}\right)\:\mathrm{a}^{\mathrm{2}} \mathrm{da}\:+\:\mathrm{2ab}\:\mathrm{db}\:+\:\mathrm{b}^{\mathrm{2}} \:\mathrm{da}\:=\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{d}\left(\mathrm{ab}^{\mathrm{2}} \right)\:+\:\mathrm{a}^{\mathrm{2}} \:\mathrm{da}\:=\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\int\:\mathrm{d}\left(\mathrm{ab}^{\mathrm{2}}…
Question Number 128664 by bemath last updated on 09/Jan/21 $$\:\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \left(\mathrm{1}−\mathrm{sin}\:\mathrm{x}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}−\mathrm{sin}\:^{\mathrm{3}} \mathrm{x}+\mathrm{sin}\:^{\mathrm{4}} \mathrm{x}−\mathrm{sin}\:^{\mathrm{5}} \mathrm{x}+…\right)\:\mathrm{dx}\:=? \\ $$ Answered by liberty last updated on 09/Jan/21…