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Category: Differential Equation

find-all-aplication-f-in-R-R-f-C-2-x-R-f-x-f-x-x-

Question Number 144608 by ArielVyny last updated on 26/Jun/21 $${find}\:{all}\:{aplication}\:{f}\:{in}\:\mathbb{R}\rightarrow\mathbb{R}\:\:{f}\in{C}^{\mathrm{2}} \\ $$$$\forall{x}\in\mathbb{R}.\:\:{f}''\left({x}\right)+{f}\left(−{x}\right)={x} \\ $$ Answered by Olaf_Thorendsen last updated on 27/Jun/21 $${f}''\left({x}\right)+{f}\left(−{x}\right)\:=\:{x}\:\:\:\left(\mathrm{1}\right) \\ $$$$\mathrm{Let}\:{u}\left({x}\right)\:=\:{f}\left({x}\right)+{x} \\…

x-t-4x-t-5y-t-y-t-4y-t-

Question Number 13316 by 433 last updated on 18/May/17 $$\begin{cases}{{x}'\left({t}\right)=\mathrm{4}{x}\left({t}\right)+\mathrm{5}{y}\left({t}\right)}\\{{y}'\left({t}\right)=\mathrm{4}{y}\left({t}\right)}\end{cases} \\ $$ Answered by ajfour last updated on 18/May/17 $${dy}=\mathrm{4}{ydt}\:\:\:\:\: \\ $$$$\int\frac{{dy}}{{y}}=\mathrm{4}\int{dt} \\ $$$$\mathrm{ln}\:\left(\frac{{y}}{{y}_{\mathrm{0}} }\right)=\mathrm{4}{t}\:\:\:\:{or}\:\:\boldsymbol{{y}}=\boldsymbol{{y}}_{\mathrm{0}}…

y-cos-x-y-cos-2-x-

Question Number 144356 by Ar Brandon last updated on 24/Jun/21 $$\mathrm{y}'+\mathrm{cos}\left(\mathrm{x}\right)\mathrm{y}=\mathrm{cos}^{\mathrm{2}} \mathrm{x} \\ $$ Answered by Olaf_Thorendsen last updated on 24/Jun/21 $${y}'+\mathrm{cos}\left({x}\right){y}\:=\:\mathrm{cos}^{\mathrm{2}} {x}\:\:\:\left(\mathrm{1}\right) \\ $$$${y}\:=\:{e}^{−\mathrm{sin}{x}}…

Question-131488

Question Number 131488 by Ahmed1hamouda last updated on 05/Feb/21 Answered by Ñï= last updated on 18/Feb/21 $${y}''+\mathrm{2}{y}'+\mathrm{5}{y}=\mathrm{6}{e}^{\mathrm{2}{x}} +{xsin}^{\mathrm{2}} {x}+{e}^{{x}} {cos}\mathrm{2}{x} \\ $$$${y}_{{p}} =\frac{\mathrm{1}}{{D}^{\mathrm{2}} +\mathrm{2}{D}+\mathrm{5}}\left(\mathrm{6}{e}^{\mathrm{2}{x}} +{xsin}^{\mathrm{2}}…

integrate-0-npi-8u-npi-1-cos2u-du-

Question Number 131452 by Engr_Jidda last updated on 04/Feb/21 $${integrate} \\ $$$$\int_{\mathrm{0}} ^{{n}\pi} \varrho^{\left(\frac{\mathrm{8}{u}}{{n}\pi}\right)} \left(\mathrm{1}−{cos}\mathrm{2}{u}\right){du} \\ $$ Answered by bramlexs22 last updated on 04/Feb/21 $${hint}\::\:{let}\:\frac{\mathrm{8}{u}}{{n}\pi}\:=\:{t}\:\Rightarrow\:\mathrm{2}{u}=\frac{{tn}\pi}{\mathrm{4}}\:;\:{du}\:=\:\frac{{n}\pi}{\mathrm{8}}\:{dt}…

x-3-16-x-2-dx-

Question Number 131453 by Engr_Jidda last updated on 04/Feb/21 $$\int\frac{{x}^{\mathrm{3}} }{\:\sqrt{\mathrm{16}−{x}^{\mathrm{2}} }}{dx} \\ $$ Answered by bramlexs22 last updated on 04/Feb/21 $$\:\sqrt{\mathrm{16}−{x}^{\mathrm{2}} }\:=\:{l}\:\Rightarrow\:{x}^{\mathrm{2}} =\mathrm{16}−{l}^{\mathrm{2}} \:;\:{x}\:{dx}\:=\:−{l}\:{dl}…

Question-65858

Question Number 65858 by ugwu Kingsley last updated on 05/Aug/19 Commented by mathmax by abdo last updated on 05/Aug/19 $${x}+\mathrm{2}{y}^{'} \:+{y}\:=\left({x}+\mathrm{2}\right)^{\mathrm{3}} \:\Rightarrow\mathrm{2}{y}^{'} \:+{y}\:=\left({x}+\mathrm{2}\right)^{\mathrm{3}} −{x}\:\Rightarrow \\…