1. ## Seperable Equations - Word Problem. Help Please.

I need help with this question below. I don't really understand the problem. I tried drawing a diagram, but it still didn't help me much. I'm especially confused on the part where it states that "S satisfies a first-order differential equation", and how do I "solve it"?

Any help will be truly appreciated.

2. The equation in $\displaystyle T$ is $\displaystyle \dots$

$\displaystyle \frac{d^{2}T}{dr^{2}}+\frac{2}{r} \frac{dT}{dr}=0; T(1)=15,T(2)=25$ (1)

Setting $\displaystyle \frac{dT}{dr}=S$ the (1) becomes…

$\displaystyle \frac {dS}{dr}=-\frac {2}{r} \cdot S$ (2)

… which can be written as $\displaystyle \dots$

$\displaystyle \frac {dS}{S}= - \frac {2}{r} \cdot dr$ (3)

Integrating both terms of (3) we have $\displaystyle \dots$

$\displaystyle \ln S = - \int \frac {2}{r} \cdot dr = -2 \cdot \ln|r| + \ln c_{1} \rightarrow S= \frac {c_{1}} {r^{2}}$ (4)

Further integration gives us $\displaystyle \dots$

$\displaystyle T= \int \frac {c_{1}}{r^{2}}\cdot dr + c_{2}= - \frac {c_{1}}{2\cdot r} + c_{2}$ (5)

The conditions given in (1) permit us to find $\displaystyle c_{1}=40, c_{2}=35$, so that the solution is…

$\displaystyle T=35 - \frac{20}{r}$ (6)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$