Question Number 293 by 123456 last updated on 25/Jan/15 $${u}:\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R} \\ $$$$\frac{\partial{u}}{\partial{x}}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)+\mathrm{2}{xu}=\mathrm{0} \\ $$ Answered by prakash jain last updated on 19/Dec/14…
Question Number 291 by 123456 last updated on 25/Jan/15 $${a}\left({n},{m}\right)=\begin{cases}{{m}}&{{n}\leqslant\mathrm{0}}\\{{a}\left({n}−\mathrm{1},{m}+\mathrm{2}\right)}&{{n}>\mathrm{0}\wedge{n}\equiv\mathrm{0}\left(\mathrm{mod}\:\mathrm{2}\right)}\\{{a}\left({n}−\mathrm{2},{m}−\mathrm{1}\right)+{nn}}&{{n}>\mathrm{0}\wedge{n}\equiv\mathrm{1}\left(\mathrm{mod}\:\mathrm{2}\right)\wedge{m}\leqslant\mathrm{0}}\\{{a}\left({m}−\mathrm{1},{n}−\mathrm{1}\right)+{a}\left({n}−\mathrm{2},{m}−\mathrm{2}\right)}&{{n}>\mathrm{0}\wedge{n}\equiv\mathrm{1}\left(\mathrm{mod}\:\mathrm{2}\right)\wedge{m}>\mathrm{0}}\end{cases} \\ $$$$\mathrm{evaluate}\:{a}\left(\mathrm{7},\mathrm{5}\right) \\ $$ Answered by prakash jain last updated on 19/Dec/14 $${a}\left(\mathrm{7},\mathrm{5}\right)={a}\left(\mathrm{4},\mathrm{6}\right)+{a}\left(\mathrm{5},\mathrm{3}\right) \\ $$$$={a}\left(\mathrm{3},\mathrm{8}\right)+{a}\left(\mathrm{2},\mathrm{4}\right)+{a}\left(\mathrm{3},\mathrm{1}\right)…
Question Number 65827 by ~ À ® @ 237 ~ last updated on 04/Aug/19 $${Prove}\:{that}\:\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\int_{\frac{\mathrm{1}}{\mathrm{6}}} ^{\frac{\mathrm{5}}{\mathrm{6}}} \:\:\frac{{dv}}{\left(\mathrm{1}−\:^{{v}} \sqrt{{u}}\:\right)^{{v}} }\right){du}={ln}\mathrm{2}−\mathrm{2}{ln}\left(\sqrt{\mathrm{3}}−\mathrm{1}\right) \\ $$…
Question Number 131363 by shaker last updated on 04/Feb/21 Commented by MJS_new last updated on 04/Feb/21 $$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{think}\:\mathrm{we}\:\mathrm{can}\:\mathrm{exactly}\:\mathrm{solve}\:\mathrm{this}.\:\mathrm{we} \\ $$$$\mathrm{need}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{x}^{\mathrm{3}} +\mathrm{3}{x}+\mathrm{1}\:\mathrm{and}\:\mathrm{in}\:\mathrm{the}\:\mathrm{next} \\ $$$$\mathrm{step}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{a}\:\mathrm{polynome}\:\mathrm{of}\:\mathrm{degree}\:\mathrm{4}\:\mathrm{if} \\ $$$$\mathrm{we}\:\mathrm{are}\:\mathrm{smart}…\:\mathrm{otherwise}\:\mathrm{degree}\:\mathrm{6} \\…
Question Number 289 by 123456 last updated on 25/Jan/15 $${x}^{\mathrm{3}} \frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }+{x}^{\mathrm{2}} \frac{{dy}}{{dx}}+{xy}=\mathrm{2}\left({x}^{\mathrm{2}} −\mathrm{1}\right) \\ $$$${y}\left(\mathrm{1}\right)=\mathrm{0} \\ $$$${y}'\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=\mathrm{5} \\ $$ Answered by prakash jain…
Question Number 65825 by ~ À ® @ 237 ~ last updated on 04/Aug/19 $${let}\:{consider}\:\:{two}\:{real}\:{numbers}\:{p}\:{and}\:{such}\:{as}\:{p}^{\mathrm{2}} −{q}^{\mathrm{2}} ={pq} \\ $$$${Prove}\:{that} \\ $$$$\:\:\:{J}=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dv}}{\:^{{q}} \sqrt{\left(\mathrm{1}+\:^{{q}} \sqrt{{v}^{{p}}…
Question Number 281 by samarth last updated on 25/Jan/15 $$\mathrm{If}\:{f}\left({x}\right)=\mathrm{tan}^{−\mathrm{1}} \sqrt{\frac{\mathrm{1}+\mathrm{sin}\:{x}}{\mathrm{1}−\mathrm{sin}\:{x}}},\:\mathrm{0}\leqslant{x}\leqslant\pi/\mathrm{2},\:\mathrm{then}\:{f}\:'\left(\pi/\mathrm{6}\right)=? \\ $$ Answered by 123456 last updated on 18/Dec/14 $${f}\left({x}\right)=\mathrm{tan}^{−\mathrm{1}} \sqrt{\frac{\mathrm{1}+\mathrm{sin}\:{x}}{\mathrm{1}−\mathrm{sin}\:{x}}} \\ $$$$\frac{\partial{f}}{\partial{x}}=\frac{\partial}{\partial{x}}\left(\mathrm{tan}^{−\mathrm{1}} \sqrt{\frac{\mathrm{1}+\mathrm{sin}\:{x}}{\mathrm{1}−\mathrm{sin}\:{x}}}\right)…
Question Number 280 by arnav last updated on 25/Jan/15 $$\mathrm{Evaluate}\:\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{cos}\:{x}−\mathrm{sin}\:{x}}{\left(\pi/\mathrm{4}−{x}\right)\left(\mathrm{cos}\:{x}+\mathrm{sin}\:{x}\right)} \\ $$ Answered by 123456 last updated on 18/Dec/14 $$\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{cos}\:{x}−\mathrm{sin}\:{x}}{\left(\frac{\pi}{\mathrm{4}}−{x}\right)\left(\mathrm{cos}\:{x}+\mathrm{sin}\:{x}\right)}\rightarrow\frac{\mathrm{0}}{\mathrm{0}} \\ $$$$=\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{−\mathrm{sin}\:{x}−\mathrm{cos}\:{x}}{−\left(\mathrm{cos}\:{x}+\mathrm{sin}\:{x}\right)+\left(\frac{\pi}{\mathrm{4}}−{x}\right)\left(−\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right)}…
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Question Number 279 by amandeep last updated on 25/Jan/15 $$\mathrm{If}\:{P}\:\left({x},{y}\right),\:{F}_{\mathrm{1}} =\left(\mathrm{3},\mathrm{0}\right),\:{F}_{\mathrm{2}} =\left(−\mathrm{3},\mathrm{0}\right)\:\mathrm{and}\: \\ $$$$\mathrm{16}{x}^{\mathrm{2}} +\mathrm{25}{y}^{\mathrm{2}} =\mathrm{400}\:\mathrm{then}\:{PF}_{\mathrm{1}} +{PF}_{\mathrm{2}} =? \\ $$ Answered by 123456 last updated…