7.3.3Can I separate them?

Solving Separable Differential Equations

7-125.

For each of the following differential equations, determine the particular solution $y = f(x)$ with the initial condition $f(1) = 2$ .

$\frac { d y } { d x }= 3x^2(y - 3)$

$\frac { d y } { d x }= xy^2$

$\frac { d y } { d x } = \frac { 2 y - 2 } { x ^ { 2 } }$ , $x ≠ 0$

7-126.

TEA REX

Scientists use carbon dating to help determine the age of fossils. All living creatures absorb Carbon-14, a radioactive substance. After death, Carbon-14 stops being absorbed and the amount in the body begins to decay. When a bone is dug up centuries later, scientists measure the amount of Carbon-14 remaining. They can then determine how long ago the animal was alive. It is known that the rate at which Carbon-14 decays is proportional to the amount of Carbon-14 remaining in the bone at any given time. It is also known that the half-life of Carbon-14 is $5730$ years. This means that after $5730$ years $50\%$ of the original amount remains.

Setup a differential equation that models the rate of decay of Carbon-14 in the fossil. Use $k$ as the constant of proportionality and $y$ as the amount of Carbon-14 remaining.
Use implicit integration to determine the solution to the differential equation and solve for the constant of proportionality, $k$ .
Rex is having a tea party in his backyard with his sister when something catches his eye in the garden. After a little digging he unearths a large bone. Convinced this was a dinosaur bone, he has it carbon-dated and learns that the bone contained $92\%$ of its original amount of Carbon-14. How old is Rex’s bone?

7-127.

IT’S CONTAGIOUS!

“Did you hear about the virus that is being spread throughout the senior class?” Martin asks Henry.

“Yeah! This time of the year is when it is the most contagious!” Henry exclaims.

Martin and Henry wonder if this situation can be modeled using a differential equation. They let $S(t)$ represent the number of senior students who have contracted the virus at time $t$ minutes where $t ≥ 0$ . Through observation, they discover that $S(t)$ is increasing at a rate proportional to $80 − 2S$ .

If three seniors initially come to school with the virus, write an equation for $S(t)$ in terms of $k$ and $t$ , where $k$ is the constant of proportionality.
If seven seniors are infected ten minutes later, solve for $k$ to write an equation for $S(t)$ in terms of $t$ .
To the nearest student, how many seniors have contracted the virus after $30$ minutes?
Calculate $\lim\limits _ { t \rightarrow \infty } S ( t )$ and interpret its meaning in context.
How many senior students have contracted the virus when it is spreading the fastest? Justify your answer.

7-128.

Use implicit integration to solve the differential equation $\frac { d y } { d x }= \cos^2(y)$ .

7-129.

Determine the particular solution to the differential equation $\frac { d y } { d x }=\frac { - 3 y } { x }$ given the initial condition $f(1) = -1$ and state its domain.

7-130.

Integrate. Homework Help ✎

$\int \cos(x) \sin^2(x)dx$

$\int −\cos(3x − 2)dx$

$\int x^{-4}e^{x^{-3}}dx$

$\int ( e ^ { 7 x - 2 } - \sqrt { 5 x } ) d x$

$\int 2 \ln(4x)dx$

$\int \frac { 9 } { 9 + x } d x$

7-131.

Locate all relative extrema and points of inflection for $f(x) = 2x^{5/3} - 5x^{4/3}$ . Homework Help ✎

7-132.

Evaluate each limit. Homework Help ✎

$\lim\limits _ { x \rightarrow \infty } \frac { x } { \sqrt { x ^ { 2 } + 2 } }$

$\lim\limits _ { x \rightarrow - \infty } \frac { x } { \sqrt { x ^ { 2 } + 2 } }$

$\lim\limits _ { x \rightarrow 2 } \frac { \frac { 1 } { x } - \frac { 1 } { 2 } } { x ^ { 2 } - 4 }$

7-133.

Compute the following derivatives. Homework Help ✎

$\frac { d } { d x } [ \operatorname { sin } ^ { 3 } ( x ) ]$

$\frac { d } { d k } [ \sqrt [ 3 ] { 7 + 4 k - 2 k ^ { 2 } } ]$

$\frac { d } { d x } [ \pi ^ { x } x ^ { \pi } ]$

7-134.

Regina’s position along a straight wooden bridge is given by $x(t) = \log(t)$ , where $x$ is measured in feet and $t$ is measured in seconds. At $t = 1$ she passes her friend Angela who is going the other way, and then at $t = 10$ she stops to gaze at the water below. Homework Help ✎

Calculate Regina’s average velocity over the interval $1 ≤ t ≤ 10$ .
At what time in this interval was Regina traveling at her average velocity?

7-135.

Set up an equation to represent the following situation. Be sure to define your variables.

The rate at which a bucket leaks water, in $\text{cm}^3/\text{min}$ , is directly proportional to the amount of water in the bucket at any given time. Homework Help ✎

7-136.

Multiple Choice: Evaluate $\int _ { 0 } ^ { 400 } ( 5 ^ { x } - 2 ^ { x } ) d x + \int _ { 1 } ^ { 400 } ( 2 ^ { x } - 5 ^ { x } ) d x$ . Homework Help ✎

$0.000$

$≈ 1.043$

$≈ 0.221$

$≈ 1.820$

$∞$