Solving Tough Differential Equations with Parameters: A Step-by-Step Approach

[juice34]

eq#1) -1/r(d/dr)(r*tao)=0
eq#2)tao=m(-dv/dr)^n (n is a parameter)

for v=0 at r=R and v=V and r=kR (k and V are parameters)

I have no idea how to start this or what is the correct way to start.

I initially integrated tao to get dtao/dr=m(-dv/dr)^n-1, but this just complicates things further with the n-1. Next i plugged in tao into the first equation and then used the product rule so -1/r(d/dr)(r*m(-dv/dr)^n)=0 once again i get the n-1 factor. Then i tried taking the derivative of #1 with respect to r to yield -(1*tao)/r=0. Please could someone please guide me in the right steps!

 

[HallsofIvy]

So your equations are:
-\frac{1}{r}\frac{dr\tau}{dr}= 0
and
\tau= m\left(\frac{dv}{dr}\right)^n ?

Well, multiplying by -r, the first equation is just \frac{d r\tau}{dr}= 0 so that r\tau= constant and
\tau= \frac{C}{r}
Then the second equation becomes
\frac{C}{r}= m\left(\frac{dv}{dr}\right)^n
so
\left(\frac{dv}{dr}\right)^n= \frac{C}{mr}
\frac{dv}{dr}= \frac{C^{1/n}}{m^{1/n}r^{1/n}}
dv= \left(\frac{C}{m}\right)^{1/n} r^{-1/n}dr
and integrate.

 

[juice34]

Thank you Hallsofivey!