Question

Exercise. Use Abel's theorem to find the general solultion. Note in each problem one solution is given. (a) t2y"- 4ty' +6y = 0, where t> 0, using that yi (t) t2 is a solution. (b) y" 2y +y 0, using that y t)= e-t is a solution (c) ty"-(4t 2t2)// + (6+ 4t)y = 0, where t > 0, using that y1(t) = t2 is a solution

Answer 1

Guan dilleranded epnchan Spudt whansluan Pㅋ류 J- ce =CE ,4 =ct 4 Guen thut 9-67 integradng factv F) = e 1 General Doluthan epuchan ankyration 기+구P ㅜ. Contant EL 1c 20LBE General holuch an euaan D +k)EcE 7 = AE8E wlu A=1te 8-c

+2 +9 =0 C4t piely'+ley=o) PLE)=2 Bby Able's thesrem Spienlt Wlrovsleroan lC, ce -2dt -26 = ce -24- = Ce 2 ce 2 Ce nterady fact ) = esidt Genaral noludi an 4eputhan ke .Gengral olutan 4 quhan -t =1+K)ettc6et y=Aette wla A=1+1

Guen dolevandal euahan y(46+2) p lE7= -Spuedt WCY,, Ce .:By Ali's Theorem ce RA- 42 4=26 -2logt loge PF= e = e Genaral Aoluhan epachan 24 2 Genarel Deluhon d enustan D 2 =9+9=+ 2 dl A= tK