Here is a problem i've been working on. I think i've got the first two parts but I'm stuck on the last section. I'd appreciate anyone checking my solution so far and even better helping me with the third section.

Thanks in advance!

Q: A particle of massmkg moves in a horizontal straight line from the originOwith initial velocityUi, whereiis the unit vector in the direction of motion. A resistive forceiacts on the particle, where is a constant andiis the velocity of the particle at time seconds measured from the start of the motion.

(i) Show that the velocity of the particle satisfies the differential equation

,

where is the distance of the particle from .

Hence show that .

(ii) Using (i) or otherwise, show that

.

(iii) Find an expression, in terms ofkandU, for the time taken for the speed of the particle to reduce to half its initial value.

Solution:

(i)

as required

then

but at

so

and

as required

(ii)

but at

so

and

so as required.

(iii) Help please!