Answer
The order is 3.
It is linear.
Work Step by Step
The differential equation $$\frac{d^3y}{dt^3}+t\frac{dy}{dt}+(\cos^2t)y=t^3$$ has order 3 because the highest derivative appearing in the equation, $\frac{d^3y}{dt^3}$, is of order 3.
To see it is linear, we subtract $t^3$ from both sides to get $$\frac{d^3y}{dt^3}+t\frac{dy}{dt}+(\cos^2t)y-t^3=0.$$ Since the left hand side is a linear function of $y$ and its derivatives, our original differential equation is linear.
