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## Homework Statement

A particle falling under gravity is subjet to a retarding force proportional to its velocity. Find its position as a function of time, if it starts from rest, and show that it will eventually reach a terminal velocity.

**2. The attempt at a solution**

To save myself some latex time, I skipped a couple of the intermediary steps here:

[tex]F_{tot}=F_{g}-F_{retarding}[/tex]

[tex]m\ddot{x}=-mg-\lambda\dot{x}[/tex]

[tex]\ddot{x}=-g-\frac{\lambda}{m}\dot{x}[/tex]

let

[tex]\gamma=\frac{\lambda}{m}[/tex]

then

[tex]\frac{d}{dt}dx=-gdt-\gamma dx[/tex]

integrating both sides

[tex]\frac{d}{dt}\int_{o}^{x}dx=-g\int_{0}^{t}dt-\gamma\int_{0}^{x}dx[/tex]

[tex]\dot{x}+\gamma x=-gt[/tex]

This final equation I get is correct (hopefully my procedure is too) and at this point, the question offers a hint:

[HINT: After the first integration, use an integrating factor, i.e. a function f(t) such when the equation is multiplied by f(t), the left-hand side becomes an exact derivative, in fact, the derivative of xf. The final stage requires an integration by parts]

This is where I got stuck. How do I determine the integrating factor f(t)? And what exactly is meant by "the left-hand side becomes an exact derivative, in fact, the derivative of xf"?

Your help is appreciated, thanks!

phyz