IVP differential equations

Question

How would you go about solving a differential equations IVP where

dy/dt= ty - t/y , y(0)=3

Answer 1

Its not quite a bad as it looks: when you separate the variables, you get the following equation

dy/(y - 1/y) = t dt or y dy/(y² - 1) = t dt to obtain

(1/2)ln(y² - 1) = t²/2 + Constant or ln(y² - 1) = t² + Constant

or y² - 1 = Ke^t2 (K = Constant) then y = SQRT(Ke^t2 + 1)

To satisfy y(0) = 3 means that K = 8 and y = SQRT(8e^t2 + 1)