Using Separation of Variables to Modify Neutron Density Diff

[tasm]

I was overlooking page 47 of "The Physics of the Manhattan Project" 2.2 Critical Mass: Diffusion Theory, and author Bruce Cameron Reed reported that:

$\Big (\frac{\partial N}{\partial t} \Big)_t = \frac{v}{\lambda_f} \big( \nu-1 \big) N + \frac{v \lambda_t}{3} \big( \nabla^2 N_r \big)_r$ \hfill (2.18)

can be modified. Basically, since $N(r,t) = N(t)_tN(r)_r$ we can use separation of variables to modify (2.18) to get $\frac{1}{N_t} \Big (\frac{\partial N_t}{\partial t} \Big)_t = \frac{v}{\lambda_f} \big( \nu-1 \big) + \frac{v \lambda_t}{3 N_r} \Bigg( \frac{1}{r^2} \frac{\partial}{\partial r} \Bigg( r^2 \frac{\partial N_r}{\partial r} \Bigg) \Bigg)$ \hfill (2.19)

Can anyone explain how Bruce Cameron Reed got from (2.18) to (2.19)

I tried plugging $N(r,t) = N(r) N(t)$ into (2.18) to get (2.19), but it just does not make any sense to me on how there is a fraction of $\frac{1}{N_r}$ multiplied to the second term after the equals sign

Assuming $N$ in the first term after the equals sign is $N(r,t)$ I cannot see how using algebra would allow a person to arrive to (2.19)

I have also tried to take advantage of the relationship

$\Big (\frac{\partial N(r,t)}{\partial t} \Big)_t = N'(t) N(r)$ but even taking advantage of this I still could not figure out how Reed transformed (2.18) to (2.19)

 

[tasm]

I was overlooking page 47 of "The Physics of the Manhattan Project" 2.2 Critical Mass: Diffusion Theory, and author Bruce Cameron Reed reported that:
Can anyone explain how Bruce Cameron Reed got from (2.18) to (2.19)

I tried plugging $N(r,t) = N(r) N(t)$ into (2.18) to get (2.19), but it just does not make any sense to me on how there is a fraction of $\frac{1}{N_r}$ multiplied to the second term after the equals sign

Assuming $N$ in the first term after the equals sign is $N(r,t)$ I cannot see how using algebra would allow a person to arrive to (2.19)

I have also tried to take advantage of the relationship

$\Big (\frac{\partial N(r,t)}{\partial t} \Big)_t = N'(t) N(r)$ but even taking advantage of this I still could not figure out how Reed transformed (2.18) to (2.19)

BY THE WAY, I FORGOT TO GIVE A LINK TO THE PDF OF THE BOOK. HERE IT IS:

http://download.springer.com/static/pdf/164/bok%253A978-3-642-14709-8.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-3-642-14709-8&token2=exp=1477589641~acl=%2Fstatic%2Fpdf%2F164%2Fbok%25253A978-3-642-14709-8.pdf%3ForiginUrl%3Dhttp%253A%252F%252Flink.springer.com%252Fbook%252F10.1007%252F978-3-642-14709-8*~hmac=43061743582b27e324f2238b065b3a347a0078d2a4cc041f0781d563e9739e2a