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If you're looking for the number of dogs (I'm assuming, you didn't specify), all you'd have to do is divide the number by two, then round down to find the number of dogs, then check your answer by adding that number and that number plus 5.
For example, yours would look like 125/6=62.5, and rounding down to 60.
Then check your answer by adding 60 and 65, the number of cats. 60+65=125.
There are 60 dogs inside the kennel.
0.6n = n + 31.8
0.6n - n = 31.8
-0.4n = 31.8
n = 31.8 / -0.4
n = -79.5
Answer:
![A = \frac{P}{r}\left( e^{rt} -1 \right)](https://tex.z-dn.net/?f=A%20%3D%20%5Cfrac%7BP%7D%7Br%7D%5Cleft%28%20e%5E%7Brt%7D%20-1%20%5Cright%29)
Step-by-step explanation:
This is <em>a separable differential equation</em>. Rearranging terms in the equation gives
![\frac{dA}{rA+P} = dt](https://tex.z-dn.net/?f=%5Cfrac%7BdA%7D%7BrA%2BP%7D%20%3D%20dt)
Integration on both sides gives
![\int \frac{dA}{rA+P} = \int dt](https://tex.z-dn.net/?f=%5Cint%20%5Cfrac%7BdA%7D%7BrA%2BP%7D%20%3D%20%5Cint%20%20dt)
where
is a constant of integration.
The steps for solving the integral on the right hand side are presented below.
![\int \frac{dA}{rA+P} = \begin{vmatrix} rA+P = m \implies rdA = dm\end{vmatrix} \\\\\phantom{\int \frac{dA}{rA+P} } = \int \frac{1}{m} \frac{1}{r} \, dm \\\\\phantom{\int \frac{dA}{rA+P} } = \frac{1}{r} \int \frac{1}{m} \, dm\\\\\phantom{\int \frac{dA}{rA+P} } = \frac{1}{r} \ln |m| + c \\\\&\phantom{\int \frac{dA}{rA+P} } = \frac{1}{r} \ln |rA+P| +c](https://tex.z-dn.net/?f=%5Cint%20%5Cfrac%7BdA%7D%7BrA%2BP%7D%20%3D%20%5Cbegin%7Bvmatrix%7D%20rA%2BP%20%3D%20m%20%5Cimplies%20rdA%20%3D%20dm%5Cend%7Bvmatrix%7D%20%5C%5C%5C%5C%5Cphantom%7B%5Cint%20%5Cfrac%7BdA%7D%7BrA%2BP%7D%20%7D%20%3D%20%5Cint%20%5Cfrac%7B1%7D%7Bm%7D%20%5Cfrac%7B1%7D%7Br%7D%20%5C%2C%20dm%20%5C%5C%5C%5C%5Cphantom%7B%5Cint%20%5Cfrac%7BdA%7D%7BrA%2BP%7D%20%7D%20%3D%20%5Cfrac%7B1%7D%7Br%7D%20%5Cint%20%5Cfrac%7B1%7D%7Bm%7D%20%5C%2C%20dm%5C%5C%5C%5C%5Cphantom%7B%5Cint%20%5Cfrac%7BdA%7D%7BrA%2BP%7D%20%7D%20%3D%20%5Cfrac%7B1%7D%7Br%7D%20%5Cln%20%7Cm%7C%20%2B%20c%20%5C%5C%5C%5C%26%5Cphantom%7B%5Cint%20%5Cfrac%7BdA%7D%7BrA%2BP%7D%20%7D%20%3D%20%5Cfrac%7B1%7D%7Br%7D%20%5Cln%20%7CrA%2BP%7C%20%2Bc)
Therefore,
![\frac{1}{r} \ln |rA+P| = t+c](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Br%7D%20%5Cln%20%7CrA%2BP%7C%20%3D%20t%2Bc)
Multiply both sides by ![r.](https://tex.z-dn.net/?f=r.)
![\ln |rA+P| = rt+c_1, \quad c_1 := rc](https://tex.z-dn.net/?f=%5Cln%20%7CrA%2BP%7C%20%3D%20rt%2Bc_1%2C%20%5Cquad%20c_1%20%3A%3D%20rc)
By taking exponents, we obtain
![e^{\ln |rA+P|} = e^{rt+c_1} \implies |rA+P| = e^{rt} \cdot e^{c_1} rA+P = Ce^{rt}, \quad C:= \pm e^{c_1}](https://tex.z-dn.net/?f=e%5E%7B%5Cln%20%7CrA%2BP%7C%7D%20%3D%20e%5E%7Brt%2Bc_1%7D%20%5Cimplies%20%20%7CrA%2BP%7C%20%3D%20e%5E%7Brt%7D%20%5Ccdot%20e%5E%7Bc_1%7D%20rA%2BP%20%3D%20Ce%5E%7Brt%7D%2C%20%5Cquad%20C%3A%3D%20%5Cpm%20e%5E%7Bc_1%7D)
Isolate
.
![rA+P = Ce^{rt} \implies rA = Ce^{rt} - P \implies A = \frac{C}{r}e^{rt} - \frac{P}{r}](https://tex.z-dn.net/?f=rA%2BP%20%3D%20Ce%5E%7Brt%7D%20%5Cimplies%20rA%20%3D%20Ce%5E%7Brt%7D%20-%20P%20%5Cimplies%20A%20%3D%20%5Cfrac%7BC%7D%7Br%7De%5E%7Brt%7D%20-%20%5Cfrac%7BP%7D%7Br%7D)
Since
when
, we obtain an initial condition
.
We can use it to find the numeric value of the constant
.
Substituting
for
and
in the equation gives
![0 = \frac{C}{r}e^{0} - \frac{P}{r} \implies \frac{P}{r} = \frac{C}{r} \implies C=P](https://tex.z-dn.net/?f=0%20%3D%20%5Cfrac%7BC%7D%7Br%7De%5E%7B0%7D%20-%20%5Cfrac%7BP%7D%7Br%7D%20%5Cimplies%20%5Cfrac%7BP%7D%7Br%7D%20%3D%20%5Cfrac%7BC%7D%7Br%7D%20%5Cimplies%20C%3DP)
Therefore, the solution of the given differential equation is
![A = \frac{P}{r}e^{rt} - \frac{P}{r} = \frac{P}{r}\left( e^{rt} -1 \right)](https://tex.z-dn.net/?f=A%20%3D%20%5Cfrac%7BP%7D%7Br%7De%5E%7Brt%7D%20-%20%5Cfrac%7BP%7D%7Br%7D%20%3D%20%5Cfrac%7BP%7D%7Br%7D%5Cleft%28%20e%5E%7Brt%7D%20-1%20%5Cright%29)