Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 30/en

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Warm-up-exercises

Exercise

Sketch the underlying vector fields of the differential equations

as well as the solution curves given in Example 30.6, Example 30.7 and Example 30.8.


Exercise

Confirm by derivation that the curves we have found in Example 30.6, Example 30.7 and Example 30.8 are the solution curves of the differential equations


Exercise

Interpret a location-independent differential equation as a differential equations with separable variables using the theorem for differential equations with separable variables.


Exercise

Determine all the solutions to the differential equation

using the theorem for differential equations with separable variables.


Exercise

Determine all the solutions to the differential equation

using the theorem for differential equations with separable variables.


Exercise

Determine all the solutions to the differential equation

using the theorem for differential equations with separable variables.


Exercise

Solve the differential equation

using the theorem for differential equations with separable variables.


Exercise

Consider the solutions

to the logistic differential equation we have found in Example 30.9.

a) Sketch up the graph of this function (for suitable and ).

b) Determine the limits for and .

c) Study the monotony behavior of these functions.

d) For which does the derivative of have a maximum (For the function itself, this represents an inflection point).

e) Which symmetries have these functions?


Exercise

Find a solution for the ordinary differential equation

with and .


Exercise

Determine the solutions for the differential equation ()

using separation of variables. Where are the solutions defined?




Hand-in-exercises

Exercise (3 points)

Prove that a differential equation of the shape

with a continuous function

on an interval has the solution

where is an antiderivative of such that .


Exercise (3 points)

Determine all the solutions to the differential equation

using the theorem for differential equations with separable variables.


Exercise (4 points)

Determine all the solutions to the differential equation

using the theorem for differential equations with separable variables.


Exercise (5 points)

Determine the solutions to the differential equation

by using the approach for

a) inhomogeneous linear equations,

b) separated variables.




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