Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 28/en/latex

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\zwischenueberschrift{Warm-up-exercises}




\inputexercise
{}
{

Let
\mathl{x \in \R}{} and consider the function \maabbeledisp {f} {\R_+} {\R } {t} { f(t) = t^x e^{-t} } {.} Determine the extrema of this function.

}
{} {}




\inputexercise
{}
{

Prove that for the factorial function the relationship
\mathdisp {\operatorname{Fak} \, { \left({ \frac{ 2k-1 }{ 2 } }\right) } = { \frac{ \prod_{i = 1}^{k} (2i-1) }{ 2^k } } \cdot \sqrt{\pi}} { }
holds.

}
{} {}




\inputexercise
{}
{

a) Prove that for
\mathl{x \geq 1}{} the estimate
\mavergleichskettedisp
{\vergleichskette
{ \int_{ 1 }^{ \infty } t^x e^{-t} \, d t }
{ \leq} {1 }
{ } { }
{ } { }
{ } { }
} {}{}{} holds.

b) Prove that the function
\mathl{H(x)}{} defined by
\mathdisp {H(x) = \int_{ 1 }^{ \infty } t^x e^{-t} \, d t} { }
for
\mathl{x \geq 1}{} is increasing.

c) Prove that
\mathl{10! \geq e^{11} +1}{.}

d) Prove that for the factorial function for
\mathl{x \geq 10}{} the estimate
\mathdisp {\operatorname{Fak} \, (x) \geq e^x} { }
holds.

}
{} {}




\inputexercise
{}
{

Solve the initial value problem
\mathdisp {y'= \sin t \text{ with } y(\pi) = 7} { . }

}
{} {}




\inputexercise
{}
{

Solve the initial value problem
\mathdisp {y'= 3t^2-4t+7 \text{ with } y(2) = 5} { . }

}
{} {}




\inputexercise
{}
{

Find all the solutions for the ordinary differential equation
\mathdisp {y'=y} { . }

}
{} {}




\inputexercise
{}
{

Make clear and mathematically clear to yourself that in a location-independent differential equation (i.e. $f(t,y)$ does not depend on $y$) the difference between two solutions \mathkor {} {y_1} {and} {y_2} {} does not depend on time, that is
\mathl{y_1(t)- y_2(t)}{} is constant. Show with an example that this may not happen in a time-independent differential equation.

}
{} {}






\zwischenueberschrift{Hand-in-exercises}




\inputexercise
{2}
{

Prove that for the factorial function the relationship
\mathdisp {\operatorname{Fak} \, (x) = \int_{ 0 }^{ 1 } (- \ln t)^x \, d t} { }
holds.

}
{} {}




\inputexercise
{3}
{

Solve the initial value problem
\mathdisp {y'= 3t^3-2t+5 \text{ with } y(3) = 4} { . }

}
{} {}




\inputexercise
{3}
{

Find a solution for the ordinary differential equation
\mathdisp {y'=t + y} { . }

}
{} {}




\inputexercise
{4}
{

Solve the initial value problem
\mathdisp {y'= { \frac{ t^3 }{ t^2+1 } } \text{ with } y(1) = 2} { . }

}
{} {}



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