Need some help with differential equations for mechanics.

[uber_kim]

Homework Statement

I'm having problems with some differential equations, just need to know where I'm going wrong.

Homework Equations

The Attempt at a Solution

a) mv\stackrel{dv}{dx}=F(x)
mvdv=F(x)dx
m∫vdv=∫F(x)dx
v^2=vv^{2}_{0}+\stackrel{2}{m}∫F(x)dx

Setting F(x)=-kx
v^2=v^{2}_{0}-\stackrel{k}{m}(x^2-x^{2}_{0})

I then have to find the position as a function of time..
\stackrel{dx}{dt}=\sqrt{v^{2}_{0}-\stackrel{k}{m}(x^2-x^{2}_{0}}
dx/\sqrt{v^{2}_{0}-\stackrel{k}{m}(x^2-x^{2}_{0}}=dt

I'm not sure how to do that integral, though, or if that's even right.

b) This problem involves a disc moving along a rough surface, so it has friction (F) and linear air resistance (-bv) acting on it.

ma=-bv+F
mdv/dt=-bv+F
-\stackrel{m}{b}∫dv/v=∫Fdt
-\stackrel{m}{b}ln(\stackrel{v}{v_0}=Ft
e^(-\stackrel{m}{b})v/v_o=e^(Ft)
v=v_0e^(-Fbt/m)

Thanks for any help!

 

[voko]

For a), transform the integrand to this form: $\displaystyle \frac {a} {\sqrt {1 - (cx)^2 } }$, then use the substitution $u = cx$.

For b), you are not doing it correctly. You should have gotten $\displaystyle \int \frac {dv} {F - bv}$.