## Elementary Differential Equations and Boundary Value Problems 9th Edition

The differential equation $$\frac{d^4y}{dt^4} +\frac{d^3y}{dt^3}+\frac{d^2y}{dt^2} +\frac{dy}{dt}+y=1$$ has order 4 because the fourth derivative $\frac{d^4y}{dt^4}$ is the highest derivative in the equation. To see it is linear, we subtract $1$ from both sides to get $$\frac{d^4y}{dt^4} +\frac{d^3y}{dt^3}+\frac{d^2y}{dt^2} +\frac{dy}{dt}+y-1=0.$$ Since the left hand side is a linear function of $y$ and its derivatives, our original differential equation is linear.