Question

Question 2 Given the following second order IVP: y" – 2y' + y = e*, y(0) = 0, y'(0) = 1. 1. Solve it using the undetermined coefficients method. [2 pt] 2. Solve it again using the Laplace Transform. (2.5 pt] 3. Which method did you find easier and why? [0.5 pt]

Answer 1

Given y"-24' + y = ex > To Solve of (1) y=en moram + 1 = 0 å yle) = 0, y'lo)= 1. o the differential equation besing the undetermined Coefficients me flod. To find CF Let 80 be trivial solation of homogeous Then the auxtarly auxiliary egh ie (m= 9) = 0 > me 1.1 CE (+2) To find PI. 1 is a double Obserne that RoHS of DEU is en and root of the auxiliary equation, hence we to take Yp = Arren [ by the rule of undetermine & coefficienty method ] TA' = 2 Arena Arren Aneren § Y = 242"+ 2aren it ex en DEW 11. DEC DE a Differentral Antena 4 Arena 2Aen_ 4Amen - 2 Amen Amen en = 2aenzen Now putting them in Heu he ] equat 2A 1 Thas the geneoal solution CF + YP. y= y = 6*262) 4 2 2 2 At hifral condition, y(o)=0 xoo yao Now from (9) , we get OF G H 0.ceeon Lotº » c 0 At y'Cole! Dietas. vido a Y' & Gea Gena a Cena aantrer putting go notoylad os et go to 40 = 1

Therefore the required Laplace Toansforma 4of you? = Lefen and Step a: Taking Caplace doansform of both sides A(S-1) + BC Sale required Solutron y = xebeuren 2 / To Save the DE o ewing of DECU. We have Afy2;-24413 Sher Lofemśng. Denote 4f ylus 3 boy yees. { Y"]= sayes) - syros - () L {y'} = syls) - y101. Arplaying the initial condition ycol = 0!Co) 7.1 , they become. 4 4hy!}= sy(s)-1 444'} = syees Equation " becomes s'yis)- 1 - 28 y 15) + y(s) - istana => 1896 25 +19 Yes) = ( 25 +1) Y(5) (3) Step 2: We now solve equal sua tron (3) for yess -1) (25+1) tepsi me Li's (3-1)3 S-vv + -1) TS-1) 1 Itsal sol S (S-1) We have at once yricson S 14 are most now determine S Now, S (S-1) an S (S-13 A(5425+1) of BS Bac => S = A$^++-24+B)S +(A. B+C) Equating Coefficreati prs.so Aro - 24+ Be 1, A-B* CEO we ger

hence Bad, c=1 :. S (S-13 13 - CS-1) oleja 1 2 (5-gary (S-1) 4176-1)99 Life 2:4418-ayou? anoan ren 2002 + 4 27 from (A), Ylx) = 4 a yea) = (S -)) nen terren which is re solution of DE D. & The easier method is andetermined coefficient method. because andetermined coefficient methad is less Calculation than Laplace transform. required