Mathematics, 23.09.2019 12:00 henrry9091
Find an implicit and an explicit solution of the given initial-value problem. (use x for x( dx dt = 2(x2 + 1), x(π/4) = 1
Answers: 1
![\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]](/tpl/images/0254/7791/37505.png)
, considering that 
:![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}}](/tpl/images/0254/7791/561e7.png)
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Find an implicit and an explicit solution of the given initial-value problem. (use x for x( dx dt =...
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