Tag: Applications

Discuss properties of the solution of a differential equation

by RoRi

February 24, 2016

Consider the differential equation

$y' = \frac{2y^2 + x}{3y^2 + 5}.$

Let $y = f(x)$ be a solution to the equation with the initial condition $f(0)= 0$ . Without attempting to solve this equation explicitly answer the following questions.

Since $f(0)= 0$ we also have $f'(0) = 0$ . Does $f$ have a relative maximum, relative minimum, or neither at 0?
If $x \geq 0$ then $f'(x) \geq 0$ and if $x \geq \frac{10}{3}$ then $f'(x) \geq \frac{2}{3}$ . Find positive numbers $a$ and $b$ such that
$f(x) > ax - b \qquad \text{for each } x \geq \frac{10}{3}.$
Prove that
$\lim_{x \to +\infty} \frac{x}{y^2} = 0.$
Prove that
$\lim_{x \to +\infty} \frac{y}{x} = A$

for some finite number $A$ and find the value of $A$ .

Incomplete.

Find the launch velocity necessary for a rocket to escape Earth only considering gravity

by RoRi

February 24, 2016

Ignoring all forces except the Earth’s gravity find the speed with which you must launch a rocket so that it never returns to Earth.

Incomplete.

Find and solve a differential equation governing population growth with given conditions

by RoRi

February 24, 2016

The population of a town at time $t = 0$ is 365. The population growth factor is $e$ , and the town experiences a death rate of one percent of the population per day. Find a differential equation modeling the population of the town as a function of time and find

the actual population of the town after $t$ years,
the cumulative total of the fatalities from the town’s death rate.

Incomplete.

Find and solve a differential equation for the decay of a material with given properties

by RoRi

February 24, 2016

A given substance decays at a rate proportional to the squareroot of the amount of the material present. Show that a substance with this decay property will completely decay in a finite amount of time and find this time.

Incomplete.

Find and solve a differential equation for the decay of a material with given properties

by RoRi

February 24, 2016

A given substance decays at a rate proportional to the square of the amount of the material present. At the end of one year there is 0.5 grams of the substance remaining.

Create and solve a differential equation that governs the mass of the material present after $t$ years.
Find the decay constant of the material in units $\operatorname{gm}^{-1} \operatorname{yr}^{-1}$ .

Incomplete.

Find the time until a tank with a hole is empty

by RoRi

February 23, 2016

Consider a water tank shaped like a right circular cone with its point up. Find the amount of time necessary for the tank to completely drain if there is a hole in its base. Express the time as a function of the dimension of the water tank and the area $A_0$ of the hole in the tank.

Incomplete.

Find the depth of water in a leaky tank as time goes to infinity

by RoRi

February 23, 2016

Consider a water tank with vertical sides and with cross-section a square of area 4 feet. Water exits the tank through a hole which as area equal to \frac{5}{3} square inches, and water is added to the tank at a rate of 100 cubic inches per second. Show that the water level approaches the value of $\left( \frac{25}{24} \right)^2$ feet above the hole no matter what the initial water level was.

Incomplete.

Find a function whose ordinate set generates a solid of revolution with volume x² f(x)

by RoRi

February 23, 2016

Let $f(x)$ be a nonnegative, differentiable function whose graph passes through both points $(0,0)$ and $\left( 1, \frac{2}{\pi}\right)$ . For every real number $x > 0$ , the ordinate set of $f$ on the interval $[0,x]$ generates a solid of revolution when rotated about the $x$ -axis whose volume is given by

$x^2 f(x).$

Find the formula for the function $f(x)$ .

Incomplete.

Find a function which divides a rectangle into pieces with given properties

by RoRi

February 23, 2016

Consider a curve whose Cartesian equation is given by $y = f(x)$ , and which passes through the origin. A rectangular region is drawn with one corner at the origin, and the other corner on the curve of the graph of $f(x)$ . The curve $f(x)$ then divides the rectangle into two pieces $A$ and $B$ . These two pieces of the rectangle then generate solids of revolution when rotated about the $x$ -axis. If the volume of one solid of revolution is always $n$ times the volume of the other solid of revolution, find the equation for $f(x)$ .

Incomplete.

Find a function which divides a rectangle into pieces with given properties

by RoRi

February 23, 2016

Consider a curve whose Cartesian equation is given by $y = f(x)$ , and which passes through the origin. A rectangular region is drawn with one corner at the origin, and the other corner on the curve of the graph of $f(x)$ . The curve $f(x)$ then divides the rectangle into two pieces $A$ and $B$ such that one of the regions has area $n$ times the area of the other for every such rectangle. Find the equation of $f(x)$ .

Incomplete.