§7.1 An introduction to differential equations

Introduction

What is a differential equation?

Differential equations in the world around us

An object's velocity is $v(t) = 4t+1$ feet per second. The net change in its position over $[0,4]$ equals

A town's population grows continuously at an annual rate of $1.25\%$. With $P$ the population and $t$ the years, the differential equation is

A radioactive sample loses $5.6\%$ of its mass every day. With $m$ the mass and $t$ the days, the differential equation is

A bank account earns $4\%$ interest per year continuously while \1000$per year is withdrawn continually. If$B(t)$ is the balance, the governing differential equation is

A skydiver's velocity graph yields slope readings $dv/dt = 1.25, 0.75, 0.5, 0.25$ at $v = 0.5, 1.5, 2, 2.5$. Plotting $dv/dt$ against $v$ reveals

Combining the skydiver and meteorite data, the line relating $\frac{dv}{dt}$ to $v$ is

For the skydiver model $\frac{dv}{dt} = -0.5v+1.5$, find the terminal velocity by solving $-0.5v+1.5 = 0$.

Hot chocolate at temperature $H$ in a $70^\circ$ room obeys $\frac{dH}{dt} = -0.1(H-70)$. Compute $\frac{dH}{dt}$ when $H = 90$, in degrees per minute.

A skydiver's velocity obeys $\frac{dv}{dt} = -0.5v + 1.5$. The velocity increases when

For the meteorite falling through the atmosphere, the measured values of $\frac{dv}{dt}$ are negative at every plotted velocity, meaning it slows toward its terminal velocity from above.

A cold soda warming by $10\%$ of its gap to a $70^\circ$ room per minute and a hot drink cooling by $10\%$ of its gap per minute are governed by the same differential equation $\frac{dH}{dt} = -0.1(H-70)$.

In the model $\frac{dv}{dt} = -0.5v+1.5$, every velocity above $3$ decreases with time, which is why the meteorite's speeds within the plotted window all fall.