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

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

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

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

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

${}y'=y\,,$

using the theorem for differential equations with separable variables.

Exercise

Determine all the solutions to the differential equation

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

using the theorem for differential equations with separable variables.

Exercise

Determine all the solutions to the differential equation

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

using the theorem for differential equations with separable variables.

Exercise

Solve the differential equation

${}y'=ty\,$

using the theorem for differential equations with separable variables.

Exercise

Consider the solutions

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

to the logistic differential equation we have found in an example.

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

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

c) Study the monotony behavior of these functions.

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

Exercise

Find a solution for the ordinary differential equation

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

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

Exercise

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

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

using separation of variables. Where are the solutions defined?

Hand-in-exercises

Exercise

Prove that a differential equation of the shape

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

with a continuous function

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

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

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

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

Exercise

Determine all the solutions to the differential equation

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

using the theorem for differential equations with separable variables.

Exercise

Determine all the solutions to the differential equation

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

using the theorem for differential equations with separable variables.

Exercise

Determine the solutions to the differential equation

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

by using the approach for

a) inhomogeneous linear equations,

b) separable variables.