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Lecture 2

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Kenya Levy
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28 views4 pages

Lecture 2

Uploaded by

Kenya Levy
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Solution : first of all wotice that rf shy ©& ey is really a solutiou of the given IVP, thn we should require xX>o This would imply that the dowain of He given equation is represented by Ha shaded regious heve below a Yh ie i i origin Sig} most be eludes 1% —_ The initial couditiou ycey=t is sabisfied since Ye): Ine > Is the > Is Ihe which vs Ye) f always true ! Perhaps the Siua plest way fo ver: fy that (3) is the weplics t solution of the given LvP 15 fo derive the deff. eq. satisfied by () aud coupare 16 with (). let us apply the same method as in exauple 9. Then, yo lnyy > Y= lx - Iny Afferentae Ws, dy. 1 1 dy with respect tox 3X Ydx (15) > d¥ ly > xytidys ydx dx xX (Yt) cross multiply > Ydx-x (yen dy <0 whic comades with i), We couclude that : y= ny isa solution of the Jiveu Ive. Exauple Il: we waut to discuss the solution of each of the Arffereutial equatrous 2 I) (2) tx ° dx I) (2 + y'so Solution : 4) it has no solution! Rewrite it as yk then you Tum ediately realize that squares of real-valued quoutities caunot be ragahve ! for thoe of you more motheuaticatly inclined we could arguwent as follows: T) could be veri fred by a fuucou y sf dg. geo aud K=0 (6) but then YX) = constouk dud Siuce x=0 we must have Yo) = coustout | hot is yo) is a function defined ouly at He pout Xoo where if taxes some coustaut value, Ou the other side, the appearana of 2 fu the given equation requsires Hust Suck derivative exists or iu other words that yay fs du fferewkiable. But cut uy differentiable function must be continuous aud Hur is not the case for our caudidele Y(o) z coustout suce itis defined ouly at one port | 7) this equation cau be satis fred by requiring, dy . Eoier You so Now Wo > YH)= ¢ coustaut dx . Y@lso D> Cz0 Hence, the, ouly solubiou is the trial solutou Ya)eo. Rewark : the 2° Aff. eq- iu He exauple above illustrates au excephiou to the general rele stating that the number of arbitrary coustauts ix te geveral solution of a at. eq. 1S the same as Hee order of He cue hou. Fudeed, I) is a I order deff. 09. but cts general soduhou Yod=0 couferius no arbitrary coustauts at all. this (17) exception is due to the fact that eq. I) fs nonlinear. Def. : An ordinary differential equation. dy "y\ = F(x, Hi Lys ° is linear if Fis a linear functou of the variables yay dy dy Cee) aK? dt? 7 sxe or iu other wordls if Foau be writteuas a liuear tq. combination of the variables (rk). Heuce, the general linear orolimary Aff. eg. of order n is . ts awd’ acid : ; OTe OEE tba mid saedy= gts) (4 Cxawple : 3 * 2 4 ow) JY 420 dy + XY =x 2 order linear dx3 dx? 3 2 nd @) dy, ze“d¥ i yd¥. x! 2” order hou hiuear dx3 dxz tex Source of houlinean fy! t) equation for te augle 6 of ax oscillating peudeeliua of leng Hy L SLOOLS 2 ge + % et O=0 2”. order hou linear j \ i “D soure of toes itor i; pn $ -- Gravitational acceleratiou, | ng. (18)

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