find the general solution of the following first order differential equation.

Question

u(t): du/dt = u /t+2t

pls show all working

Answer 1

I assume you mean
du/dt = (u/t) + 2t and not
u/(t + 2t) = u/(3t)

Your D.E. is of the form
du/dt + Pu = Q
where P = -1/t and Q = 2t

The method of "integrating factors" can be used. Your textbook should have an explanation of the method.

There is a function rho(t) given by
rho(t) = exp(integral of P(t)dt)

such that the solution is
rho(t)*u
= integral of (rho*Q) + C