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

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\setcounter{section}{30}






\zwischenueberschrift{Warm-up-exercises}




\inputexercise
{}
{

Sketch the underlying vector fields of the differential equations
\mathdisp {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.

}
{} {}




\inputexercise
{}
{

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

}
{} {}




\inputexercise
{}
{

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

}
{} {}




\inputexercise
{}
{

Determine all the solutions to the differential equation
\mathdisp {y'=y} { , }
using the theorem for differential equations with separable variables.

}
{} {}




\inputexercise
{}
{

Determine all the solutions to the differential equation
\mathdisp {y'= e^y} { , }
using the theorem for differential equations with separable variables.

}
{} {}




\inputexercise
{}
{

Determine all the solutions to the differential equation
\mathdisp {y'= { \frac{ 1 }{ \sin y } }} { , }
using the theorem for differential equations with separable variables.

}
{} {}




\inputexercise
{}
{

Solve the differential equation
\mathdisp {y'=ty} { }
using the theorem for differential equations with separable variables.

}
{} {}




\inputexercise
{}
{

Consider the solutions
\mathdisp {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 \zusatzklammer {for suitable \mathkor {} {s} {and} {g} {}} {} {.}

b) Determine the limits for \mathkor {} {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 \zusatzklammer {For the function itself, this represents an \stichwort {inflection point} {}} {} {.}

e) Which symmetries have these functions?

}
{} {}




\inputexercise
{}
{

Find a solution for the ordinary differential equation
\mathdisp {y' = { \frac{ t }{ t^2-1 } } y^2} { }
with \mathkor {} {t>1} {and} {y<0} {.}

}
{} {}




\inputexercise
{}
{

Determine the solutions for the \definitionsverweis {differential equation}{}{} \zusatzklammer {\mathlk{y>0}{}} {} {}
\mathdisp {y'=t^2y^3} { }
using separation of variables. Where are the solutions defined?

}
{} {}






\zwischenueberschrift{Hand-in-exercises}




\inputexercise
{3}
{

Prove that a differential equation of the shape
\mathdisp {y'= g(t)\cdot y^2} { }
with a continuous function \maabbeledisp {g} {\R} {\R } {t} {g(t) } {,} on an interval $I'$ has the solution
\mathdisp {y(t) = - \frac{1}{G(t)}} { , }
where
\mathl{G}{} is an antiderivative of $g$ such that
\mathl{G(I') \subseteq \R_+}{.}

}
{} {}




\inputexercise
{3}
{

Determine all the solutions to the differential equation
\mathdisp {y'=ty^2,\, y> 0} { , }
using the theorem for differential equations with separable variables.

}
{} {}




\inputexercise
{4}
{

Determine all the solutions to the differential equation
\mathdisp {y'=t^3y^3, \, y > 0} { , }
using the theorem for differential equations with separable variables.

}
{} {}




\inputexercise
{5}
{

Determine the solutions to the \definitionsverweis {differential equation}{}{}
\mathdisp {y'=ty + t} { }
by using the approach for

a) inhomogeneous linear equations,

b) separated variables.

}
{} {}



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