solve initial value problem

Question

t dy/dt = y + sqrt(t^2 + y^2) , y(1)=0

Answer 1

ty' = y+√(t^2+y^2)

let y=vt
then y' = v + tv'

Now we have

t(v+tv') = tv+√(t^2+(tv)^2)
tv + t^2v' = tv + t√(1+v^2)
t^2v' = t√(1+v^2)
tv' = √(1+v^2)
dv/√(1+v^2) = dt/t
arcsinh(v) = lnt + c
v = sinh(c+lnt)
y= vt = t*sinh(c+lnt)

whew