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- Calculus volume 2
- Introduction to differential
- Separable equations
A cake is removed from the oven after baking thoroughly, and the temperature of the oven is
The temperature of the kitchen is
and after
minutes the temperature of the cake is
- Write the appropriate initial-value problem to describe this situation.
- Solve the initial-value problem for
- How long will it take until the temperature of the cake is within
of room temperature?
- Initial-value problem
-
- Approximately
minutes.
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Key concepts
- A separable differential equation is any equation that can be written in the form
- The method of separation of variables is used to find the general solution to a separable differential equation.
Key equations
-
Separable differential equation
-
Solution concentration
-
Newton’s law of cooling
Solve the following initial-value problems with the initial condition
and graph the solution.
Find the general solution to the differential equation.
Find the solution to the initial-value problem.
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?
Most
drugs in the bloodstream decay according to the equation
where
is the concentration of the drug in the bloodstream. If the half-life of a drug is
hours, what fraction of the initial dose remains after
hours?
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A drug is administered intravenously to a patient at a rate
mg/h and is cleared from the body at a rate proportional to the amount of drug still present in the body,
Set up and solve the differential equation, assuming there is no drug initially present in the body.
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[T] How often should a drug be taken if its dose is
mg, it is cleared at a rate
mg/h, and
mg is required to be in the bloodstream at all times?
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A tank contains
kilogram of salt dissolved in
liters of water. A salt solution of
kg salt/L is pumped into the tank at a rate of
L/min and is drained at the same rate. Solve for the salt concentration at time
Assume the tank is well mixed.
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A tank containing
kilograms of salt dissolved in
liters of water has two salt solutions pumped in. The first solution of
kg salt/L is pumped in at a rate of
L/min and the second solution of
kg salt/L is pumped in at a rate of
L/min. The tank drains at
L/min. Assume the tank is well mixed. Solve for the salt concentration at time
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Questions & Answers
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Source:
OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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