4.5 First-order linear equations  (Page 7/10)

[T] Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall $5000$ meters if the mass is $100$ kilograms, the acceleration due to gravity is $9.8{\text{m/s}}^2$ and the proportionality constant is $4?$ Does it take more or less time than your initial estimate?

Solve the generic equation $y\prime =ax+y.$ How does varying $a$ change the behavior?

Solve the generic equation $y\prime =ax+y.$ How does varying $a$ change the behavior?

Solve the generic equation $y\prime =ax+xy.$ How does varying $a$ change the behavior?

Solve the generic equation $y\prime =x+axy.$ How does varying $a$ change the behavior?

Solve $y\prime -y=e^{kt}$ with the initial condition $y(0)=0.$ As $k$ approaches $1,$ what happens to your formula?

The differential equation $y\prime =3x^{2}y-\text{cos}(x)y\text{″}$ is linear.

The differential equation $y\prime =x-y$ is separable.

You can explicitly solve all first-order differential equations by separation or by the method of integrating factors.

You can determine the behavior of all first-order differential equations using directional fields or Euler’s method.

$y^{\prime }=x^2+3e^x-2x$

$y\prime =2^x+{\text{cos}}^{\mathrm{-1}}x$

$y\prime =y\left(x^2+1\right)$

$y\prime =e^{\text{−}y}\text{sin}x$

$y\prime =3x-2y$

$y\prime =y\text{ln}y$

$y\prime =8x-\text{ln}x-3x^4,y(1)=5$

$y\prime =3x-\text{cos}x+2,y(0)=4$

$xy\prime =y\left(x-2\right),y(1)=3$

$y\prime =3y^2\left(x+\text{cos}x\right),y(0)=\mathrm{-2}$

$(x-1)y\prime =y-2,y(0)=0$

$y\prime =3y-x+6x^2,y(0)=\mathrm{-1}$

$y\prime =2y-y^2$

$y\prime =\frac{1}{x}+\text{ln}x-y,$ for $x>0$

$y\prime =\mathrm{-4}yx,y(0)=1$

$y\prime =3^x-2y,y(0)=0$

A car drives along a freeway, accelerating according to $a=5\text{sin}(\pi t),$ where $t$ represents time in minutes. Find the velocity at any time $t,$ assuming the car starts with an initial speed of $60$ mph.

You throw a ball of mass $2$ kilograms into the air with an upward velocity of $8$ m/s. Find exactly the time the ball will remain in the air, assuming that gravity is given by $g=9.8{\text{m/s}}^2.$

You drop a ball with a mass of $5$ kilograms out an airplane window at a height of $5000$ m. How long does it take for the ball to reach the ground?

You drop the same ball of mass $5$ kilograms out of the same airplane window at the same height, except this time you assume a drag force proportional to the ball’s velocity, using a proportionality constant of $3$ and the ball reaches terminal velocity. Solve for the distance fallen as a function of time. How long does it take the ball to reach the ground?

A drug is administered to a patient every $24$ hours and is cleared at a rate proportional to the amount of drug left in the body, with proportionality constant $0.2.$ If the patient needs a baseline level of $5$ mg to be in the bloodstream at all times, how large should the dose be?

A $1000$ -liter tank contains pure water and a solution of $0.2$ kg salt/L is pumped into the tank at a rate of $1$ L/min and is drained at the same rate. Solve for total amount of salt in the tank at time $t.$

You boil water to make tea. When you pour the water into your teapot, the temperature is $100\text{°C.}$ After $5$ minutes in your $15\text{°C}$ room, the temperature of the tea is $85\text{°C.}$ Solve the equation to determine the temperatures of the tea at time $t.$ How long must you wait until the tea is at a drinkable temperature $(72\text{°C})?$

The human population (in thousands) of Nevada in $1950$ was roughly $160.$ If the carrying capacity is estimated at $10$ million individuals, and assuming a growth rate of $2\text{\%}$ per year, develop a logistic growth model and solve for the population in Nevada at any time (use $1950$ as time = 0). What population does your model predict for $2000?$ How close is your prediction to the true value of $\mathrm{1,998,257}?$

Repeat the previous problem but use Gompertz growth model. Which is more accurate?