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!