Solve the differential equation $\dfrac{dy}{dx}=x^{2}+2x+1$ with the boundary condition when $x=3,$ then $y={-}1.$

Solution We begin by finding the general solution of the differential equation, namely $$y=\frac{x^{3}}{3}+x^{2}+x+C$$

To determine the number $C$, we use the boundary condition when $x=3,$ then $y={-}1$. $$\begin{array}{rl@{\qquad}l} -1 &=\dfrac{3^{3}}{3}+3^{2}+3+C & \color{#0066A7}{{x=3,\quad y=-1}} \\[8pt] C&=-22 \end{array}$$

The particular solution of the differential equation with the boundary condition when $x=3,$ then $y=-1$, is $$y=\frac{x^{3}}{3}+x^{2}+x-22$$