calculus Initial value problem

Question

solve the Initial value problem for y=y(x) where

xyy’ = (x^2)+1

y(1) =2

Answer 1

This is a separable differential equation which can be solved as follows:

xy(dy/dx) = x² + 1

ydy = (x² + 1) / x dx

ydy = (x + 1/x) dx Now integrate both sides:

1/2y² = 1/2x² + ln|x| + C Use intitial condition to solve for C:

1/2 (2)² = 1/2 (1)² + ln|1| + C

2 = 1/2 + C

C = 3/2

1/2y² = 1/2x² + ln|x| + 3/2 Isolate y:

y² = x² + 2ln|x| + 3

y = √(x² + 2ln|x| + 3)