Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 30
- Warm-up-exercises
Exercise
Sketch the underlying vector fields of the differential equations
as well as the solution curves given in [[|the examples]].
Exercise
Confirm by derivation that the curves we have found in the examples 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 an example.
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).
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
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
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 the solutions to the differential equation
by using the approach for
a) inhomogeneous linear equations,
b) separable variables.