Find the general solution of the equation.

Question

6t*dy/dt+y=sqrt (t),t>0.

Answer 1

6t dy/dt + y = √t
y' + 1/6t y = 1/(6√t)

You want an integrating factor:

e^(∫1/(6t) dt) = t^(1/6)

Now the equation becomes

t^(1/6) y' + 1/6 t^(-5/6) y = 1/(6∛t)
d(t^(1/6) y) = 1/(6∛t)
t^(1/6) y = 1/4 t^(2/3) + c
y = 1/4 √t + c/t^(1/6)