From the acceleration yo can find the rest of the expressions by integrating the initial equations
[tex]\begin{gathered} a(t)=pt^2-qt^3 \\ v(t)=\int a(t)dt=\frac{pt^3}{3}-\frac{qt^4}{4}+c \\ d(t)=\int\int a(t)dtdt=\frac{pt^4}{12}-\frac{qt^5}{20}+ct+b \end{gathered}[/tex]Is important to put the constants when you integrate, they can affect the result when you solve the system
[tex]\begin{gathered} v(0)=0=\frac{p(0)^3}{3}-\frac{q(0)^4}{4}+c \\ 0=c \\ d(0)=\frac{p(0)^4}{12}-\frac{q(0)^5}{20}+b=0 \\ 0=b \end{gathered}[/tex]After having the expressions, you replace the assumptions, in this case, v and d are 0, when time t is 0
