Circumference = 2 * PI * radius
Circumference = 2 * 3.14159265 * 63
Circumference = 395.841
3/47 is a rational number
p/q is a rational number, where p and q are real integers, and q does not equal to 0.
Answer:
= 4sqrt3
Step-by-step explanation:
-2i sqrt -12
= -2i * i sqrt12 <-- i^2 = -1
= -2i^2sqrt12
= -2(-1)sqrt12
= 2sqrt12
= 2 * 2 sqrt3
= 4sqrt3
Substitute
, so that
. Then the ODE is equivalent to
![\dfrac{\mathrm dv}{\mathrm dx}+2=v^2-7](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dv%7D%7B%5Cmathrm%20dx%7D%2B2%3Dv%5E2-7)
which is separable as
![\dfrac{\mathrm dv}{v^2-9}=\mathrm dx](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dv%7D%7Bv%5E2-9%7D%3D%5Cmathrm%20dx)
Split the left side into partial fractions,
![\dfrac1{v^2-9}=\dfrac16\left(\dfrac1{v-3}-\dfrac1{v+3}\right)](https://tex.z-dn.net/?f=%5Cdfrac1%7Bv%5E2-9%7D%3D%5Cdfrac16%5Cleft%28%5Cdfrac1%7Bv-3%7D-%5Cdfrac1%7Bv%2B3%7D%5Cright%29)
so that integrating both sides is trivial and we get
![\dfrac{\ln|v-3|-\ln|v+3|}6=x+C](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cln%7Cv-3%7C-%5Cln%7Cv%2B3%7C%7D6%3Dx%2BC)
![\ln\left|\dfrac{v-3}{v+3}\right|=6x+C](https://tex.z-dn.net/?f=%5Cln%5Cleft%7C%5Cdfrac%7Bv-3%7D%7Bv%2B3%7D%5Cright%7C%3D6x%2BC)
![\dfrac{v-3}{v+3}=Ce^{6x}](https://tex.z-dn.net/?f=%5Cdfrac%7Bv-3%7D%7Bv%2B3%7D%3DCe%5E%7B6x%7D)
![\dfrac{v+3-6}{v+3}=1-\dfrac6{v+3}=Ce^{6x}](https://tex.z-dn.net/?f=%5Cdfrac%7Bv%2B3-6%7D%7Bv%2B3%7D%3D1-%5Cdfrac6%7Bv%2B3%7D%3DCe%5E%7B6x%7D)
![\dfrac6{v+3}=1-Ce^{6x}](https://tex.z-dn.net/?f=%5Cdfrac6%7Bv%2B3%7D%3D1-Ce%5E%7B6x%7D)
![v=\dfrac6{1-Ce^{6x}}-3](https://tex.z-dn.net/?f=v%3D%5Cdfrac6%7B1-Ce%5E%7B6x%7D%7D-3)
![-2x+y=\dfrac6{1-Ce^{6x}}-3](https://tex.z-dn.net/?f=-2x%2By%3D%5Cdfrac6%7B1-Ce%5E%7B6x%7D%7D-3)
![y=2x+\dfrac6{1-Ce^{6x}}-3](https://tex.z-dn.net/?f=y%3D2x%2B%5Cdfrac6%7B1-Ce%5E%7B6x%7D%7D-3)
Given the initial condition
, we find
![0=\dfrac6{1-C}-3\implies C=-1](https://tex.z-dn.net/?f=0%3D%5Cdfrac6%7B1-C%7D-3%5Cimplies%20C%3D-1)
so that the ODE has the particular solution,
![\boxed{y=2x+\dfrac6{1+e^{6x}}-3}](https://tex.z-dn.net/?f=%5Cboxed%7By%3D2x%2B%5Cdfrac6%7B1%2Be%5E%7B6x%7D%7D-3%7D)