Use separation of variables to solve a differential equation.
Solve applications using separation of variables.
We now examine a solution technique for finding exact solutions to a class of differential equations known as separable differential equations. These equations are common in a wide variety of disciplines, including physics, chemistry, and engineering. We illustrate a few applications at the end of the section.
Separation of variables
We start with a definition and some examples.
Definition
A
separable differential equation is any equation that can be written in the form
The term ‘separable’ refers to the fact that the right-hand side of the equation can be separated into a function of
times a function of
Examples of separable differential equations include
The second equation is separable with
and
the third equation is separable with
and
and the right-hand side of the fourth equation can be factored as
so it is separable as well. The third equation is also called an
autonomous differential equation because the right-hand side of the equation is a function of
alone. If a differential equation is separable, then it is possible to solve the equation using the method of
separation of variables .
Problem-solving strategy: separation of variables
Check for any values of
that make
These correspond to constant solutions.
Rewrite the differential equation in the form
Integrate both sides of the equation.
Solve the resulting equation for
if possible.
If an initial condition exists, substitute the appropriate values for
and
into the equation and solve for the constant.
Note that Step 4. states “Solve the resulting equation for
if possible.” It is not always possible to obtain
as an explicit function of
Quite often we have to be satisfied with finding
as an implicit function of
Using separation of variables
Find a general solution to the differential equation
using the method of separation of variables.
Follow the five-step method of separation of variables.
In this example,
and
Setting
gives
as a constant solution.
Rewrite the differential equation in the form
Integrate both sides of the equation:
Let
Then
so the equation becomes
To solve this equation for
first multiply both sides of the equation by
Now we use some logic in dealing with the constant
Since
represents an arbitrary constant,
also represents an arbitrary constant. If we call the second arbitrary constant
the equation becomes
Now exponentiate both sides of the equation (i.e., make each side of the equation the exponent for the base
Again define a new constant
(note that
This corresponds to two separate equations:
and
The solution to either equation can be written in the form
Since
it does not matter whether we use plus or minus, so the constant can actually have either sign. Furthermore, the subscript on the constant
is entirely arbitrary, and can be dropped. Therefore the solution can be written as
No initial condition is imposed, so we are finished.
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
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Shukri
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Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
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sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
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Asui
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Cornelius
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Cornelius
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The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
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D,alculate optimum level of output .
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
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