Answer
$y=\frac{x^3}{3}-x+\frac{7}{3}$
Work Step by Step
The answer to this solution relies on solving the differential equation for the particular solution. Given that:
$\frac{dy}{dx}=x^2-1$ with a particular point on it $(-1,3)$
We multiply both sides by $dx$
$dy=(x^2-1)dx$
We now integrate both sides, each with respect to the differential in each expression.
$\int{dy}=\int{(x^2-1)dx}$
$y=\frac{x^3}{3}-x+c$
The integration step above is done by applying the power rule for integration. Integrating the value 1 with respect to any variable results in the variable itself.
Solving for c, we just make use of the points given to us.
$3=\frac{(-1)^3}{3}-(-1)+c$
$c=\frac{7}{3}$
Hence, the particular solution is:
$y=\frac{x^3}{3}-x+\frac{7}{3}$
