Find solution to ivp xy' + 3y = x(sqrt(y)), y(1) = 0, bernoulli equation

1 To solve my '+3y = x1, y(1) = 0 Divide the above equation with x, we get 3 y'+-y= y2... (1) Clearly (1) is the Bernoulli's differenti al equation On multiplying (1) by we get y2 X 1 dy 3 y + = 1 x 1 + 1 = 1 dy 3 y3 =1...(2) T dx Put ył =z 1 - dy dx 1 dy v dx Put these values in (2), we get 3 +z=1 X dz 31 1 +--z=-.. .(3) dx 2 x Clearly which is linear in z and which is of the form z'+P(x)2 = Q(x) 31 Hence P(x) Q(x)= 2 x The integrating factor 4(x) = espandex يا أنا" $(x) = x?

The general solution of (3) is 24(x) = [ 4(x)Q(x) dx+c (2)=1*()to 3 x²dxto 3 1 x2 23 2 +c + = x3+ S = -x +c 5 = zx x +c 5 13 yaxa x2 +c...(4) y(1) = 0 Put x = 1, y = 0 in (4), we get 0=-+ 1 C=-- -- 5 Put c=- in (4), we get 1 3 1 = -x2 5 5 = x2 1 y= 25 } Which is the required solution of the given D.E.