미분방정식 전체 문제

If given a half-life of t years, the constant $k$ for $y={e}^{kt}$ is calculated by $k=\ln \,(1/2)/t.$

How old is a skull that contains one-fifth as much radiocarbon as a modern skull? Note that the half-life of radiocarbon is $5730$ years.

If a relic contains $90\%$ as much radiocarbon as new material, can it have come from the time of Christ (approximately $2000$ years ago)? Note that the half-life of radiocarbon is $5730$ years.

The populations of New York and Los Angeles are growing at $1\%$ and $1.4\%$ a year, respectively. Starting from $8$ million (New York) and $6$ million (Los Angeles), when are the populations equal? Round your answer to a whole number of years.

For the following exercises, use $y={y}_{0}{e}^{kt}.$

If $y=1000$ at $t=3$ and $y=3000$ at $t=4,$ what was ${y}_{0}$ at $t=0?$

For the following exercises, use $y={y}_{0}{e}^{kt}.$

If $y=100$ at $t=4$ and $y=10$ at $t=8,$ when does $y=1?$

If a bank offers annual interest of $7.5\%$ or continuous interest of $7.25\%,$ which has a better annual yield?

What continuous interest rate has the same yield as an annual rate of $9\%?$

You are an archaeologist and are given a bone that is claimed to be from a Tyrannosaurus Rex. You know these dinosaurs lived during the Cretaceous Era $(146$ million years to $65$ million years ago), and you find by radiocarbon dating that there is $0.000001\%$ the amount of radiocarbon. Is this bone from the Cretaceous?

The spent fuel of a nuclear reactor contains plutonium-239, which has a half-life of $24,000$ years. If $1$ barrel containing $10\,\text{kg}$ of plutonium-239 is sealed, how many years must pass until only $10g$ of plutonium-239 is left?

Determine the order of the following differential equations.

${y}^{'}+y=3{y}^{2}$

Determine the order of the following differential equations.

${({y}^{'})}^{2}={y}^{'}+2y$

Determine the order of the following differential equations.

$y'''+{y}^{''}{y}^{'}=3{x}^{2}$

Determine the order of the following differential equations.

${y}^{'}={y}^{''}+3{t}^{2}$

Determine the order of the following differential equations.

$\frac{dy}{dt}=t$

Determine the order of the following differential equations.

$\frac{dy}{dx}+\frac{{d}^{2}y}{d{x}^{2}}=3{x}^{4}$

Determine the order of the following differential equations.

${\left(\frac{dy}{dt}\right)}^{2}+8\frac{dy}{dt}+3y=4t$

Find the particular solution to the differential equation ${y}^{'}=4{x}^{2}$ that passes through $(-3,-30),$ given that $y=C+\frac{4{x}^{3}}{3}$ is a general solution.

Find the particular solution to the differential equation ${y}^{'}=3{x}^{3}$ that passes through $(1,4.75),$ given that $y=C+\frac{3{x}^{4}}{4}$ is a general solution.

Find the particular solution to the differential equation ${y}^{'}=3{x}^{2}y$ that passes through $(0,12),$ given that $y=C{e}^{{x}^{3}}$ is a general solution.