Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 28

Warm-up-exercises

Exercise

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

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

Determine the extrema of this function.

Exercise

Prove that for the factorial function the relationship

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

holds.

Exercise

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

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

holds.

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

${}H(x)=\int _{1}^{\infty }t^{x}e^{-t}dt\,$

for ${}x\geq 1$ is increasing.

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

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

${}\operatorname {Fac} \,x\geq e^{x}\,$

holds.

Exercise

Solve the initial value problem

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

Exercise

Solve the initial value problem

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

Exercise

Find all the solutions for the ordinary differential equation

${}y'=y\,.$

Exercise

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

Hand-in-exercises

Exercise

Prove that for the factorial function the relationship

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

holds.

Exercise

Solve the initial value problem

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

Exercise

Find a solution for the ordinary differential equation

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

Exercise

Solve the initial value problem

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