# How to integrate when zero is in denominator?

+ 0 like - 0 dislike
170 views

I have this equation:

$$(p_0 p_1)'=\dfrac{4}{r^2-1}p_0'$$

where $'$ is derivative $\dfrac{d}{dz}$.

I need to solve it (find $p_1$) for case when $r=1$ (I already know $p_0$),  and I don't know what to do because in that case I will have zero in denominator. I can make this shape:

$$(r^2-1)(p_0 p_1)'=4p_0'$$
where I will have zero on left side and just $p_0'$ on the right side, but this equation is part of the system, where I already found solution for $p_0$.

Can you tell me please do you see any solution of this?

Before upper equation it was necessary to solve this one:

$$4\beta\dfrac{\partial^2 u}{\partial r^2} + \dfrac{4\beta}{r}\dfrac{\partial u}{\partial r}=\dfrac{\partial p_1}{\partial z}$$

where $\beta$ is constant, $p=f(z)$, $r$ is radial coordinate, $z$ is longitudinal coordinate and $u=u(r,z)$ and with additional conditions:

$r=0: \dfrac{\partial u}{\partial r}=0$;

$r=1: u=0$.

I got solution: $$u(r,z)=\frac1{16\beta}\,(r^2-1)\,\frac{dp_1}{dz}(z).$$ from this equation is my part in upper equation $(r^2-1)$.

Because first equation is probably not correct because zero in denominator, can you tell me please did I make mistake in solution for $u(r,z)$?

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\varnothing$ysicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.