Time to reach a speed given the acceleration equation

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I have modeled the acceleration of a car, and here is the speed of this car with respect to the time : $$\frac{dv}{dt} = \frac{6.21}{v}-0.046-0.000137v^2$$

Now, how can I find the time needed to reach a given speed ?

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You could separate variables: $$\int_{t_1}^{t_2} dt=\int_{v_1}^{v_2} \frac{dv}{\frac{a}{v}-b-cv^2}$$

(Where $a$, $b$, and $c$ are the constants in your expression for the acceleration.)

Not sure if it is possible to find an antiderivative for the right hand side, though. As Bye_World commented, you might need to use numerical methods. Euler's method for solving ODE's is a simple one. Runge-Kutta method is slightly more complicated, but more accurate.

Edit: Nicolas pointed out in the comments that we actually can find an antiderivative for the right hand side. WolframAlpha gives (assuming $v$ is not zero): $$t_2-t_1 = \sum_{\{\omega\,:\,a-b\omega-c\omega^2=0\}}\frac{\omega \ln(\frac{v_2-\omega}{v_1-\omega})}{3c\omega^2+b}$$ The sum is over the three roots of $a-b\omega-c\omega^3$. The expression is still pretty complicated but it gives you the time interval to accelerate from $v_1$ to $v_2$, as desired.