Question

Assuming x > 0, find the general solution of the following Euler equa- tions. a) x²y" – 3xy' +4y=0 (b)x²y" – 5xy +10y=0 f) 5x2y" + 12. y' + 2y = 0 g) x²y" + xy = 0

Please do 1a 1b 1d thanks

Answer 1

Here we find general solution of Euler equation for x>0.

SolM: To find solution of Eeler eyuations tsv 70 qnd that all solutions ave of the form, J Buler eqn is aqxytbxytcyo pt y n ths eqn we get, +Cu 70 qnd so this will anly be zero if ar (Y-1)t6(y) +C 0 30 5utions wil be of the frm provided r is Ex: 8) niyu- 3xy'y= first the root of q=1,b-3, c=4 Y (v-)-3rt 4 = o 2 11 4V+40 Y-2 Y-20 2,2 the general solh is then 2 2 nt4.-5+ 4y First the root of , Y(-1-57t 10 10 6+ F4 621 6+136-40 3ti 2

The general soltion is then, 2 Cos(In ) t C2 SIn(lhn) y(=C 4) 9 1,6 0 C= / first the root of Y(T-1)t1O 2 1 t J1-4 1H3 1 J3 The gener solution is then, 12 cos{ J In a) + C2 x'2 sin( lnn) w =Gx