Mathematics in economics
Lecture 10
Mgr. Jiří Mazurek, Ph.D.
Mathematics in Economics/PMAT

Lecture 10
Geometric function series




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Geometric function series is defined as follows:
The series is convergent if             , where q = f(x).
The sum is given as:

Lecture 10
Geometric function series – Problem 1




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Find the range of convergence:          .
Solution:
Expanding the sum yields:
Clearly, a1 = q = x.
Because          , we have             . The range of convergence:

Lecture 10
Geometric function series – Problem 2




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1.
Find the sum of the series:          .
Solution:
We already know that a1 = q = x.
Using the formula for the sum yields:
This result is valid for all x satysfying

Lecture 10
Geometric function series – Problem 3




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1.
Find the range of convergence and a sum of the series:
Solution:
                               ,

The convergence:
Which yields:
The sum:

Lecture 10
Problems to solve




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Find the range of convergence and a sum of the series:

Lecture 10
Differential equations




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Differential equation (DE) is an equation that includes given function y = f(x) and its
derivatives.
Examples:
                      is a DE of the first order and degree 1.
                               is a DE of the first order and degree 2.
                                         is a DE of the second order and
   degree 3.

Lecture 10
Differential equations – Types of a solution




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DE can have three types of solutions:
• General solution
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• Particular solution
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• Singular solution

Lecture 10
Differential equations – Types of a solution
Example 1




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Find general solution of DE          and particular solution for a condition              .
General solution:
We simply integrate DE:
Particular solution for the initial condition: we substitute x = 0 and y = 2 into general solution:
Which yields C = 2. Thus, particular solution is

Lecture 10
Differential equations – Types of a solution
Example 2




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1.
Find general solution of DE               and particular solution for a condition  y (1) =
2.
General solution:
We integrate DE:
Particular solution for the initial condition: we substitute x = 1 and y = 2 into general solution:
Which yields C = - 2. Thus, particular solution is

Lecture 10
Differential equations – Types of a solution
Example 3




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1.
Find general solution of DE                 and particular solution for a conditions
and            .

 General solution:
Particular solution for the initial condition:
Which yields C1 = 0, C2 = 1. Thus, particular solution is:

Lecture 10
Differential equations – Separation of variables




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One of the most used method for solving DE is separation of variables. In this method x and y
variables are separated on the different sides of an equation before integration takes place.
It can be used when DE is separable:
                                      or

Lecture 10
Differential equations – Separation of variables
Example 1




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Find a general solution of            .
The equation is separable:              , so we separate both variables:
And integrate:
Which yields:

Lecture 10
Differential equations – Separation of variables
Example 2




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Find a general solution of                         .
The equation is separable, so we separate and integrate:

Lecture 10
Differential equations – Separation of variables
Example 3




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Find a general solution of                         .
The equation is separable, so we separate and integrate:

Lecture 10
Differential equations – Homogenous differential equations




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A DE of the form                     such that
is called homogenous differential equation. It is solved via substitution:
and                    .
Example:                     is homogenous, because:

Lecture 10
Differential equations – Homogenous differential equations – Example 1




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Find a general solution of a homogenous DE:
We start with the substitution            :

Lecture 10
Differential equations – Homogenous differential equations – Example 1 – cont.




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And at the end we integratate:
which yields:

Lecture 10
Differential equations – Logistic equation and function




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  In economics, demographics and other disciplines appears a function called a logistic function.
This function arises as a solution to the following logistic equation:
For an initial condition              the solution is:

Lecture 10
Differential equations – Logistic equation and function




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graf 45

Lecture 10
Differential equations
 Linear differential equations of the first order

By a linear differential equations of the first order we mean an equation of the form:
Assume that q(x) = 0:
This special equation is called homogenous, and is solved by separation of variables:

Lecture 10
Differential equations
 Linear differential equations of the first order – cont.

And finally we obtain:

Lecture 10
Differential equations
 Linear differential equations of the first order – Example 1

Find the general solution:                  .
Solution:
We follow the procedure from the previous slide:

Lecture 10
Linear differential equations of the first order
Problems to solve

    Find the general solution:

Lecture 10
Thank you for your attention