Visualizing the Initial Value Problem – Handwritten Notes | MAT 2705, Study notes of Differential Equations

Download Visualizing the Initial Value Problem – Handwritten Notes | MAT 2705 and more Study notes Differential Equations in PDF only on Docsity!

Visual izin the inthal value problem Ceute but nob necessary ) WP: andoder DE for y&) > gensaln YS Yrit C22 yor) = de —T? YOwI= CO Yibo) TAY) = Yo Yyitwl =Ve ° > YC) = A Yylbs) Hee H!Gy = Ve ©) Titreduce inthal data veckorss | Ye) = ao ? C2 N20 (23) = =l¥ =yY Y= Ey} (Cal 7 Coo Yabo Cel [4 Saving the imhal Cprachens ts exunvalont fo . ( 569 trying bh express the opt tarhal dal yeour Wy yajyQo in bers ol a Umea cometnahen of the incisal data 5 vedrs of the Lasts finchuns {yyy for the San spac, Tre metric of nis bicar systen is their Wringdeian meine ab KEM. There is q unique correspondence between inthel data and saluttions — foreach pare Conwy there is a ume sduttiin y so we Can Use these parameters on the space of solutions as coordinales just Wee we canuse (Q..Cz) as Wordindes— theyare releled to exch other by the above linear trans formation, Vo ple t But we dont anon advance whatthe solution funchos yp Yr) ave that correspon +o OW j_-* Cc ()9) and 41) WH the (Yo, ve) plane, We shactunt, enowny (Yi Y2) From p Yo our solhon technique. Yo Co} go rather than thinking of (Yo,3 inital date. Space ‘." space of sols ag the old cowinales and Cor, Cz) : as the new Woeiinaleg as this Picture implies we must consider COj0) as the Ad cords & Cysy)va). asthe New cov rdinaley > as in the Second diagram, |= Wea” [2] old B new once You backsubshtule these into Y, and gollect coefficients oF YoVe , you Find these Space of sduhons' =" lerhal daly space funchons (hyde) * the columns of B que the y= YoGr* Von coefhicents of yinY2 Necessary ; eee J natural basis for to express the new basis oye fetal Gos ns t Wp) bo) =f 2}