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**Hello all, I want to say thank you in advance for any and all advice on my question. My classical mechanics textbook (Marion Thornton) has been taking me through motion for a particle with retarding forces.**

The example it keeps giving is:

m dv/dt = -kmv

which can be solved for:

v = v

x = v

But out of curiosity I tried using the actual drag force equation "1/2ρCAv

##\ddot x ## + 1/2ρCA##\dot x ##

How do you solve this thing? I'm stuck since it's not the standard

x'' + ax' + bx = 0

My solution yielded:

The example it keeps giving is:

m dv/dt = -kmv

which can be solved for:

v = v

_{0}e^{-kt}andx = v

_{0}/k(1-e^{-kt})But out of curiosity I tried using the actual drag force equation "1/2ρCAv

^{2}" instead of "kmv." But I can't figure out how to solve the differential:##\ddot x ## + 1/2ρCA##\dot x ##

^{2}= 0How do you solve this thing? I'm stuck since it's not the standard

x'' + ax' + bx = 0

My solution yielded:

$$ \int\frac{\mathrm{d}\dot x }{ \dot x^2} = \frac{1}{2m}\rho CA\int \mathrm{dt} $$

**which just gives some weird thing:**

$$t = e^{\frac{1}{2m}\rho CAx} $$

**which can't be right.**

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