변수분리형 미분방정식

Find the solution to the initial-value problem.

$y'=3{x}^{2}({y}^{2}+4),y(0)=0$

Find the solution to the initial-value problem.

$y'={e}^{y}{5}^{x},y(0)=\ln (\ln (5))$

Find the solution to the initial-value problem.

$y'=-2x\tan (y),y(0)=\frac{\pi }{6}$

For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution?

[T] $y'=1-2y$

For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution?

[T] $y'={y}^{2}{x}^{3}$

For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution?

[T] $y'={y}^{3}{e}^{x}$

For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution?

[T] $y'={e}^{y}$

For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution?

[T] $y'=y\ln (x)$

Most drugs in the bloodstream decay according to the equation $y'=cy,$ where $y$ is the concentration of the drug in the bloodstream. If the half-life of a drug is $2$ hours, what fraction of the initial dose remains after $6$ hours?

A drug is administered intravenously to a patient at a rate $r$ mg/h and is cleared from the body at a rate proportional to the amount of drug still present in the body, $d$. Set up and solve the differential equation, assuming there is no drug initially present in the body.

[T] How often should a drug be taken if its dose is $3$ mg, it is cleared at a rate $c=0.1$ mg/h, and $1$ mg is required to be in the bloodstream at all times?

A tank contains $1$ kilogram of salt dissolved in $100$ liters of water. A salt solution of $0.1$ kg salt/L is pumped into the tank at a rate of $2$ L/min and is drained at the same rate. Solve for the salt concentration at time $t.$ Assume the tank is well mixed.

A tank containing $10$ kilograms of salt dissolved in $1000$ liters of water has two salt solutions pumped in. The first solution of $0.2$ kg salt/L is pumped in at a rate of $20$ L/min and the second solution of $0.05$ kg salt/L is pumped in at a rate of $5$ L/min. The tank drains at $25$ L/min. Assume the tank is well mixed. Solve for the salt concentration at time $t.$

[T] For the preceding problem, find how much salt is in the tank $1$ hour after the process begins.

Torricelli’s law states that for a water tank with a hole in the bottom that has a cross-sectional area of ${A}_{H}$ with a height of water $h$ above the bottom of the tank, the rate of change of volume of water flowing from the tank is proportional to the square root of the height of water, according to $\frac{dV}{dt}=-{A}_{H}\sqrt{2gh},$ where $g$ is the acceleration due to gravity. Note that $\frac{dV}{dt}={A}_{T}\frac{dh}{dt},$ where ${A}_{T}$ is the cross-sectional area of the tank. Solve the resulting initial-value problem for the height of water, assuming a tank of radius $24\sqrt{2}$ with a circular hole of radius $2$ ft. The initial height of water is $100$ ft.

For the preceding problem, determine how long it takes the tank to drain.

The liquid base of an ice cream has an initial temperature of $200^\circ \text{F}$ before it is placed in a freezer with a constant temperature of $0^\circ \text{F}.$ After $1$ hour, the temperature of the ice-cream base has decreased to $140^\circ \text{F}.$ Formulate and solve the initial-value problem to determine the temperature of the ice cream.

[T] The liquid base of an ice cream has an initial temperature of $210^\circ \text{F}$ before it is placed in a freezer with a constant temperature of $20^\circ \text{F}.$ After $2$ hours, the temperature of the ice-cream base has decreased to $170^\circ \text{F}.$ At what time will the ice cream be ready to eat? (Assume $30^\circ \text{F}$ is the optimal eating temperature.)

[T] You are organizing an ice cream social. The outside temperature is $80^\circ \text{F}$ and the ice cream is at $10^\circ \text{F}.$ After $10$ minutes, the ice cream temperature has risen by $10^\circ \text{F}.$ How much longer can you wait before the ice cream melts at $40^\circ \text{F?}$

You have a cup of coffee at temperature $70^\circ \text{C}$ and the ambient temperature in the room is $20^\circ \text{C}.$ Assuming a cooling rate $k\,\text{of}\,0.125,$ write and solve the differential equation to describe the temperature of the coffee with respect to time.