Question

Find the general solution: xy" – 4y' = 0

please show all work.

Answer 1

solution xyl _ uyl=0 let y=x" W y'= mom-1) Y" = m/m) 2) M-2 Put these in above differential equation a(n(n-1) 8 ) - 4(m 2-1) = 0 m(141-1) xm-7 _ um menntun (mlm-1) - um )gmt-0 me_m-um=0 m2_sm so

mesom so "m/M-5) = 0 m=0 or m=5 y = 6,20 +6245 Y(X) = 41+4725
Alternative solution is below
Here aal, 6=-4, C = 0 Solution Xyll_uyl=0 let 2= logx Now comparing with Standard equation for Cauchy fuler's method il axy" +by' +cy=0 as Then Ausillany Equation com be written m²+(b-1m +C =0 Putting back and c=0 m2+(-4-1) = 0 m²-sm zo MCM-5) = 0

0=(5-46 Jul sou ll 0 = W M) = 0,02 + کے ) OR Y(*= 6, +6,25