how do you solve the initial value problem by using separation of variables dy/dx=1+x/(sqrt of y), y(2)=9

Answer 1

Watch missing parentheses, please.

Parentheses are needed to enclose numerators and denominators, otherwise additions and subtractions will take place after the division.

Separate the variables,
dy/dx = (1+x)/sqrt(y)
sqrt(y)dy = (1+x)dx
Integrate:
(2/3)y^(3/2) = x + x²/2 + C
y = [(3/2)(x+x²/2+C)]^(2/3)
from which we can solve for C=14.
so
y(x)=[(3/2)(x+x²/2+14)]^(2/3)

Please check all arithmetic.