## Calculus 10th Edition

$y=\frac{x^3}{3}-x+\frac{7}{3}$
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: $y=\frac{x^3}{3}-x+\frac{7}{3}$