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

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Warm-up-exercises

### Exercise

Let ${\displaystyle {}x\in \mathbb {R} }$ and consider the function

${\displaystyle f\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,\,t\longmapsto f(t)=t^{x}e^{-t}.}$

Determine the extrema of this function.

### Exercise

Prove that for the factorial function the relationship

${\displaystyle \operatorname {Fak} \,{\left({\frac {2k-1}{2}}\right)}={\frac {\prod _{i=1}^{k}(2i-1)}{2^{k}}}\cdot {\sqrt {\pi }}}$

holds.

### Exercise

a) Prove that for ${\displaystyle {}x\geq 1}$ the estimate

${\displaystyle {}\int _{1}^{\infty }t^{x}e^{-t}\,dt\leq 1\,}$

holds.

b) Prove that the function ${\displaystyle {}H(x)}$ defined by

${\displaystyle H(x)=\int _{1}^{\infty }t^{x}e^{-t}\,dt}$
for ${\displaystyle {}x\geq 1}$ is increasing.

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

d) Prove that for the factorial function for ${\displaystyle {}x\geq 10}$ the estimate

${\displaystyle \operatorname {Fak} \,(x)\geq e^{x}}$

holds.

### Exercise

Solve the initial value problem

${\displaystyle y'=\sin t{\text{ with }}y(\pi )=7.}$

### Exercise

Solve the initial value problem

${\displaystyle y'=3t^{2}-4t+7{\text{ with }}y(2)=5.}$

### Exercise

Find all the solutions for the ordinary differential equation

${\displaystyle y'=y.}$

### Exercise

Make clear and mathematically clear to yourself that in a location-independent differential equation (i.e. ${\displaystyle {}f(t,y)}$ does not depend on ${\displaystyle {}y}$) the difference between two solutions ${\displaystyle {}y_{1}}$ and ${\displaystyle {}y_{2}}$ does not depend on time, that is ${\displaystyle {}y_{1}(t)-y_{2}(t)}$ is constant. Show with an example that this may not happen in a time-independent differential equation.

Hand-in-exercises

### Exercise (2 points)

Prove that for the factorial function the relationship

${\displaystyle \operatorname {Fak} \,(x)=\int _{0}^{1}(-\ln t)^{x}\,dt}$

holds.

### Exercise (3 points)

Solve the initial value problem

${\displaystyle y'=3t^{3}-2t+5{\text{ with }}y(3)=4.}$

### Exercise (3 points)

Find a solution for the ordinary differential equation

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

### Exercise (4 points)

Solve the initial value problem

${\displaystyle y'={\frac {t^{3}}{t^{2}+1}}{\text{ with }}y(1)=2.}$