Differential Equation Velocity Time

v = 10 - 0.1x^2 \\[br]\dfrac{\mathrm{d} x}{\mathrm{d} t} = 10 - 0.1x^2\\[br]\dfrac{\mathrm{d} t}{\mathrm{d} x} = \dfrac{1}{10 - 0.1x^2} = \dfrac{10}{100 - x^2}\\[br]= \dfrac{10}{(10 - x)(10 + x)} = \dfrac{1}{2} \left( \dfrac{1}{10 - x} + \dfrac{1}{10 + x} \right)

t = \dfrac{1}{2} \int \dfrac{1}{10 - x} + \dfrac{1}{10 + x} \, \mathrm{d} x\\[br]t = \dfrac{1}{2}\left( \ln(10-x) + \ln(10 + x) \right ) + c

0 = \ln(10) + c\\[br]c = -\ln(10)\\[br]t = \dfrac{1}{2}\left( \ln(10-x) + \ln(10 + x) \right ) - \ln(10)\\[br]\dfrac{1}{2}\left( \ln(10-x) + \ln(10 + x) \right ) = t + \ln(10)\\[br]\ln(10-x) + \ln(10 + x) = ln(100 - x^2) = 2(t + \ln(10))\\[br]100 - x^2 = \exp(2(t + \ln(10))) = \exp(2t + 2\ln(10)) = \exp(2t + \ln(100))\\[br]100 - x^2 = e^{2t + \ln(100)} = e^{2t}e^{\ln(100)} = 100e^{2t}\\[br]x = \sqrt{100 - 100e^{2t}} = 10\sqrt{1-e^{2t}}