Mechanics, rewrite and solve equation, change of variables.

[fed_ex]

Equation found in previous question is #: x''=gsin(a)-b(v^2)
1. Rewrite the equation # as a differential equation for v as a function of x.
2. Solve the equation to find v as a function of x.
Relevant equations: v=x'=dx/dt, x''=v'=a=dv/dt
Attempt at 1: Using the relevant equations you can rewrite # as
*: dv/dt=gsin(a) - b((x')^2)
Which i am scepticle of being correct due to my inability to solve it in the 2nd part.
Attempts at 2:Tried integrating but the whole((x')^2) confused me, i was thinking of making that (dx/dt)*(dx/dt) but got stuck again.
Tried to rewrite * but i was expecting a first order LINEAR equation, so that i could use the integrating factor method, so alas that also didn't work.


Any help would be appreciated, thankyou.

 

[tiny-tim]

hi fed_ex! welcome to pf!

(try using the X² button just above the Reply box )

dv/dt=gsin(a) - b((x')^2)

correct, but you've gone a bit too far …

try dv/dt=gsin(a) - b(v²) ​

 

[HallsofIvy]

You are asked to find a differential equation for v as a function of x so you want derivatives with respect to x, not t.
By the chain rule, x''= dv/dt= (dv/dx)(dx/dt)= v (dv/dx).

Replacing x'' in the equation by v (dv/dx) gives the equation you want.

 

[fed_ex]

Ah yes! Then separate and enjoy tasty solution.
Thanks!