 
\subsubsection {}
�e�te diferenci�ln� rovnice
$$y' = 1+y^2,\ \ y' = \frac{2y}{x},\ \ y' = \frac{-\tan x \sin^2 
y}{\cos^2 x \cot y}.$$
 



\subsubsection {}
�e�te diferenci�ln� rovnice neroz�e�en� vzhledem k $y'$:
$$y' \tan x = y,\ \ (1+e^x) y y' = e^x (\mbox{ s po��te�n� podm�nkou } 
y(0) = 1).$$
 



\subsubsection {}
S pou�it�m vhodn�ch substituc� zkuste �e�it rovnice
$$y' = (x+y)^2,\ \ y' = \frac{x+y}{x-y},\
$$
$$ \ y' =  \frac{y}{x}-1,\ \ 
(x-y) y' = x + y,$$
$$x y' = x e^{y/x}+y,\ \ y' = \frac{y}{x} + x,\ \ y' = \frac{4}{3}y + x 
\sqrt{y}.$$
 



\subsubsection {}
�e�te line�rn� rovnici  
$$y' = (\tan x) y + \cos x.$$






\subsubsection {}
Nalezn�te maxim�ln� �e�en� rovnice $y'=y$
proch�zej�c�
bodem
$(0,1).$




\subsubsection {}
Pro diferenci�ln� rovnici $yy' + xy^2 = x$
nalezn�te
(a) v�echna maxim�ln� �e�en�,\quad

(b) maxim�ln� �e�en� proch�zej�c� bodem $(1,0).$


\subsubsection {}
Pro diferenci�ln� rovnici $y' =
\frac{y^2}{x^2}$ nalezn�te

(a) v�echna maxim�ln� �e�en�, \quad
(b) maxim�ln� �e�en� proch�zej�c� bodem $(1, \frac 12),$\par
(c) v�echna maxim�ln� �e�en�, kter� jsou na sv�m defini�n�m
oboru omezen�.



\subsubsection {}
$y y'={1-2x\over y} \quad  xy'+y=y^2 $

$y'=10^{x+y} \quad \  e^{-y}(1+y')=1$

\subsubsection {}
$ y'+\sqrt{1-y^2\over 1-x^2}=0,\quad x\in
(-1,1)\quad $
 




\subsubsection {}
$y'={1+y^2\over 1+x^2},\ y(0)=1$



\subsubsection {}
$ y-xy'=b(1+x^2y'),\ y(1)=1$
\subsubsection {}
Kter� k�ivka m� tu vlastnost, �e jej� te�na
v ka�d�m bod� $S$ k�ivky  prot�n� osu $x$ v bod� $X$
a osu $y$ v bod� $Y$, p�i�em� $S$ je st�edem �se�ky $XY$.

( �e�en� $dy/dx = - y/x$, $1/x$,podle [Fraga], IV.2.14 )

\subsubsection {}
Nech� polo�as rozpadu radioaktivn� l�tky je $P$. Jak�
je "t�etino�as" ?

( �e�en� $x(t) = x(0) \exp {-kt}$,
 $x(0)/2 = x(0) \exp {-kP}$,$x(0)/3 = x(0) \exp {-kT}$,
$T = P \ln 3 / \ln 2$, podle [Fraga], I.3.8 )