Find the solution of the differential equation that satisfie

Solution

x ln x = y(1 + sqrt(8 + y^2))y\'

x ln x = y(1 + sqrt(8 + y^2))(dy/dx)

x ln x dx = y(1 + sqrt(8 + y^2)) dy

int x ln x dx = int y(1 + sqrt(8 + y^2)) dy

1/4 x² (2ln(x) - 1) + C = 1/2 y² + 1/3 (8+y²)^3/2

x=1 , y=1 -> -1/4 + C = 1/2 + 9

C = 9.75

Therefore the solution is:

1/4 x² (2ln(x) - 1) + 9.75 = 1/2 y² + 1/3 (8+y²)^3/2