Question

Let y(x) be the position of a particle at time x. Suppose that we know that the velocity of a particle satisfies the differential equation.
y'(t) = ty
with y(0)=1. We will try to give a reasonable method to approximate some of the positions of the particle.
Integrate both sides to show that
y(x)= ʃ ty(t)dt +1 from 0 to x

Answer 1

\[\frac{dy}{dt} = ty\]
\[\frac{dy}{y} = t dt\]
integrate
\[\ln y = \frac{t^{2}}{2} + C\]
\[\large y = k e^{t^{2}/2}\]