Find an implicit and an explicit solution of the given initial-value problem. (use x for x(t).) dx dt = 2(x2 + 1), x(π/4) = 1
1 answer:
![\displaystyle \dfrac{dx}{dt}=2(x^2+1)\\\\ \int_{t_0}^t dt=\int_{x_0}^x\dfrac{dx}{2(x^2+1)}\\\\ t-t_0=\dfrac{1}{2}\int_{x_0}^x\dfrac{dx}{(x^2+1)}\\\\ t-t_0=\dfrac{1}{2}\left[\arctan(x)\right]_{x_0}^x\\\\ t-t_0=\dfrac{1}{2}\left[\arctan(x)-\arctan(x_0)\right]](https://tex.z-dn.net/?f=%5Cdisplaystyle%0A%5Cdfrac%7Bdx%7D%7Bdt%7D%3D2%28x%5E2%2B1%29%5C%5C%5C%5C%0A%5Cint_%7Bt_0%7D%5Et%20dt%3D%5Cint_%7Bx_0%7D%5Ex%5Cdfrac%7Bdx%7D%7B2%28x%5E2%2B1%29%7D%5C%5C%5C%5C%0At-t_0%3D%5Cdfrac%7B1%7D%7B2%7D%5Cint_%7Bx_0%7D%5Ex%5Cdfrac%7Bdx%7D%7B%28x%5E2%2B1%29%7D%5C%5C%5C%5C%0At-t_0%3D%5Cdfrac%7B1%7D%7B2%7D%5Cleft%5B%5Carctan%28x%29%5Cright%5D_%7Bx_0%7D%5Ex%5C%5C%5C%5C%0At-t_0%3D%5Cdfrac%7B1%7D%7B2%7D%5Cleft%5B%5Carctan%28x%29-%5Carctan%28x_0%29%5Cright%5D)
We'll use
![x(\pi/4)=1](https://tex.z-dn.net/?f=x%28%5Cpi%2F4%29%3D1)
, considering that
![x_0=1, t_0=\dfrac{\pi}{4}](https://tex.z-dn.net/?f=x_0%3D1%2C%20t_0%3D%5Cdfrac%7B%5Cpi%7D%7B4%7D)
:
![t-t_0=\dfrac{1}{2}\left[\arctan(x)-\arctan(x_0)\right]\\\\ t-\dfrac{\pi}{4}=\dfrac{1}{2}\left[\arctan(x)-\arctan(1)\right]\\\\ t-\dfrac{\pi}{4}=\dfrac{1}{2}\left[\arctan(x)-\dfrac{\pi}{4}\right]\\\\ t-\dfrac{\pi}{4}=\dfrac{1}{2}\arctan(x)-\dfrac{\pi}{8}\\\\ t-\dfrac{\pi}{8}=\dfrac{1}{2}\arctan(x)\\\\ \boxed{\arctan(x)=2t-\dfrac{\pi}{4}}](https://tex.z-dn.net/?f=t-t_0%3D%5Cdfrac%7B1%7D%7B2%7D%5Cleft%5B%5Carctan%28x%29-%5Carctan%28x_0%29%5Cright%5D%5C%5C%5C%5C%0At-%5Cdfrac%7B%5Cpi%7D%7B4%7D%3D%5Cdfrac%7B1%7D%7B2%7D%5Cleft%5B%5Carctan%28x%29-%5Carctan%281%29%5Cright%5D%5C%5C%5C%5C%0At-%5Cdfrac%7B%5Cpi%7D%7B4%7D%3D%5Cdfrac%7B1%7D%7B2%7D%5Cleft%5B%5Carctan%28x%29-%5Cdfrac%7B%5Cpi%7D%7B4%7D%5Cright%5D%5C%5C%5C%5C%0At-%5Cdfrac%7B%5Cpi%7D%7B4%7D%3D%5Cdfrac%7B1%7D%7B2%7D%5Carctan%28x%29-%5Cdfrac%7B%5Cpi%7D%7B8%7D%5C%5C%5C%5C%0At-%5Cdfrac%7B%5Cpi%7D%7B8%7D%3D%5Cdfrac%7B1%7D%7B2%7D%5Carctan%28x%29%5C%5C%5C%5C%0A%5Cboxed%7B%5Carctan%28x%29%3D2t-%5Cdfrac%7B%5Cpi%7D%7B4%7D%7D)
Applying
tan in the both sides:
You might be interested in
.6
6 out of 10
six tenths
.......
1/25, easy
also, 1/25=4/100=0.04/1=0.04
A and D
By solving for x you get x=6
The answe is 12. formula is 2/3x=8
Answer:
-0.4
Step-by-step explanation:
you just take 2 of them to find it