Question

The indicated function y_(x) is a solution of the given differential equation. Use reduction of order or formula (5) in Section 4.2, e-SP(x) dx dx Y2 = Y1(x) >> (5) y? (x) as instructed, to find a second solution y2(x). x?y" + 2xy' – 6y = 0; Y=x2 Y2=

Answer 1

=) = x (1 Solution We have differential equection Xy" + 2xy - by=0 Y = 8 BRY" + play in yao , ſiz x? 2 = y + 2 y 2 d y = 0, y = 2 Compare it with y" + P(x) y' + Q(x) y = 0, we get 2 P(X)= and Q(x) = DC 29 Then of P(x) dx Y2 = Y, (x) 1 2 dx (x7) dx - ) -2 logx xa dx 14

2 =) Ya = x Lo elauxa -dx 304 > Ya = x S xa -dx 24 => yes you see dx Is = x2 dx 2) Y2 = -6+1 x x -6+1 -> Yaz Ya -5 Yaz 5x3