# 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

${\displaystyle y'={\frac {1}{y}},\,y'=ty^{3}{\text{ and }}y'=-ty^{3}}$

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

${\displaystyle y'={\frac {1}{y}},\,y'=ty^{3}{\text{ and }}y'=-ty^{3}.}$

### 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

${\displaystyle y'=y,}$

using the theorem for differential equations with separable variables.

### Exercise

Determine all the solutions to the differential equation

${\displaystyle y'=e^{y},}$

using the theorem for differential equations with separable variables.

### Exercise

Determine all the solutions to the differential equation

${\displaystyle y'={\frac {1}{\sin y}},}$

using the theorem for differential equations with separable variables.

### Exercise

Solve the differential equation

${\displaystyle y'=ty}$

using the theorem for differential equations with separable variables.

### Exercise

Consider the solutions

${\displaystyle y(t)={\frac {g}{1+\exp(-st)}}}$

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

a) Sketch up the graph of this function (for suitable ${\displaystyle {}s}$ and ${\displaystyle {}g}$).

b) Determine the limits for ${\displaystyle {}t\rightarrow \infty }$ and ${\displaystyle {}t\rightarrow -\infty }$.

c) Study the monotony behavior of these functions.

d) For which ${\displaystyle {}t}$ does the derivative of ${\displaystyle {}y(t)}$ 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

${\displaystyle y'={\frac {t}{t^{2}-1}}y^{2}}$

with ${\displaystyle {}t>1}$ and ${\displaystyle {}y<0}$.

### Exercise

Determine the solutions for the differential equation (${\displaystyle y>0}$)

${\displaystyle y'=t^{2}y^{3}}$

using separation of variables. Where are the solutions defined?

Hand-in-exercises

### Exercise (3 points)

Prove that a differential equation of the shape

${\displaystyle y'=g(t)\cdot y^{2}}$

with a continuous function

${\displaystyle g\colon \mathbb {R} \longrightarrow \mathbb {R} ,\,t\longmapsto g(t),}$

on an interval ${\displaystyle {}I'}$ has the solution

${\displaystyle y(t)=-{\frac {1}{G(t)}},}$

where ${\displaystyle {}G}$ is an antiderivative of ${\displaystyle {}g}$ such that ${\displaystyle {}G(I')\subseteq \mathbb {R} _{+}}$.

### Exercise (3 points)

Determine all the solutions to the differential equation

${\displaystyle y'=ty^{2},\,y>0,}$

using the theorem for differential equations with separable variables.

### Exercise (4 points)

Determine all the solutions to the differential equation

${\displaystyle y'=t^{3}y^{3},\,y>0,}$

using the theorem for differential equations with separable variables.

### Exercise (5 points)

Determine the solutions to the differential equation

${\displaystyle y'=ty+t}$

by using the approach for

a) inhomogeneous linear equations,

b) separated variables.