Answer
$r(t) = (\frac{k_1}{2m}~t^2+\frac{k_2~k_3}{120m^2}~t^5)~\hat{i} + (\frac{k_3}{6m}~t^3)~\hat{j}$
$v(t) = (\frac{k_1}{m}~t+\frac{k_2~k_3}{24m^2}~t^4)~\hat{i} + (\frac{k_3}{2m}~t^2)~\hat{j}$
Work Step by Step
$F_y(t) = k_3~t$
$a_y(t) = \frac{F_y(t)}{m} = \frac{k_3}{m}~t$
$v_y(t) = \frac{k_3}{2m}~t^2$
$y(t) = \frac{k_3}{6m}~t^3$
$F_x(t) = k_1+k_2~y$
$F_x(t) = k_1+k_2(\frac{k_3}{6m}~t^3)$
$a_x(t) = \frac{F_x(t)}{m} = \frac{k_1}{m}+\frac{k_2~k_3}{6m^2}~t^3$
$v_x(t) = \frac{k_1}{m}~t+\frac{k_2~k_3}{24m^2}~t^4$
$x(t) = \frac{k_1}{2m}~t^2+\frac{k_2~k_3}{120m^2}~t^5$
$r(t) = (\frac{k_1}{2m}~t^2+\frac{k_2~k_3}{120m^2}~t^5)~\hat{i} + (\frac{k_3}{6m}~t^3)~\hat{j}$
$v(t) = (\frac{k_1}{m}~t+\frac{k_2~k_3}{24m^2}~t^4)~\hat{i} + (\frac{k_3}{2m}~t^2)~\hat{j}$
