ClearAll["Global`*"];

x=2;
y=6;

ClearAll[x,y]
Integrate[(x^2),x]

?=

x1=Integrate[Sin[(x^2)],x]

?x1

?:=

x2:=Integrate[Sin[(x^2)],x]

?x2

FullForm[Hold[u=v]]

ClearAll[f1]
f1[x:_]:=x*Tanh[10Sin[x]]
f1[5]

ClearAll[f1]
f1[x_]:=x*Tanh[10Sin[x]]
f1[5]

ClearAll[f2]
f2[x_,y_]:=x*Tanh[10Sin[y]]
f2[Pi,E]

f[x]:=x*Tanh[10Sin[x]]
f[1]

ClearAll[f1]
f1[x_]:=x*Tanh[10Sin[x]]
?f1

ClearAll[f3,x]

f3[x_]=Integrate[Sin[(x^2)],x];
f3[5.]

ClearAll[f4,x]

f4[x_]:=Integrate[Sin[(x^2)],x];
f4[5.]

ClearAll[a];
a[n_]:=1.*n*a[n-1];
a[0]:=1;
?a

Trace[a[3]]

$RecursionLimit=5000;
Timing[a[3500]]

ClearAll[a];
a[n_]:=a[n]=1.*n*a[n-1];
a[0]:=1;
a[8];
Trace[a[10]]

ClearAll[f5]
f5[1,1]:="odchazim";
f5[x_,y_]:=x+y;
f5[x_,4]:=(x^2);
f5[{x_,y_}]:="nechci te";
f5[s,2]
f5[10,4]
f5[{10,4}]
f5[1,1]

(*Listy*)
ClearAll["Global`*"];

?List

list1={a,b,c,10,{4,5}};
list2={a,b,c}={2,6,4};

list2[[3]]
list2[[-1]]
list2[[4]]

??List

Table[opice,{12}]

Table[(x^i),{i,12}]

Table[(x^i),{i,3,12}]

Table[(x^i),{i,3,12,2}]

n=4;
mat=Table[(E^i)+(Pi^j),{i,n},{j,i}]

n=4;
mat=Table[(E^i)+(Pi^j),{i,n},{j,4}]

MatrixForm[mat]

mf=MatrixForm;

mat//mf

Grid[mat,Frame->All]

?->

?/.

{ucitel,jeptiska,bleska}/.ucitel->ja

x/.{x->1}

x/.{{x->1}}

x/.{{x->1},{x->2}}

dosedZaIksy

dosadZaIksy=Solve[(x^2)+5x-6==0]
4*(x^3)-5/.dosadZaIksy[[1]]
4*(x^3)-5/.dosadZaIksy[[2]]

n=4;
mat=Table[(E^i)+(Pi^j),{i,n},{j,4}]

mat/.Plus->Times

n=4;
mat=Table[((E^i)/(Pi^j)),{i,n},{j,4}]/.(E/(π^4))->pes

n=4;
mat=Table[((E^i)/(Pi^j)),{i,n},{j,4}]
mat/.((a_)/(b_)):>(b/a)

(π/E)//FullForm
((E^3)/(π^3))//FullForm

n=4;
mat=Table[((j^2)/(2^3)),{i,n},{j,4}]
mat/.((a_)/(b_)):>(b/a)

dosx[[1,1]][[2]]

dosx[[1,1,2]]

Abs[Plus@@(x/.Solve[(1+(t^2))*(x^2)+t*x-4==0,x])]

rce=(1+(t^2))*(x^2)+t*x-4==0;
dosx=Solve[rce,x];
x1=x/.dosx[[1]];
x2=x/.dosx[[2]];
Plot[Abs[x1+x2],{t,0,6Pi}]

(*zde druhy prednes pocina*)
ClearAll["Global`*"];

n=4;
mat=Table[((E^i)/(Pi^j)),{i,n},{j,4}]
mat/.a_:>(1/a)
mat/.a_:>((Denominator[a])/(Numerator[a]))

ClearAll[f];
f[x_]:=1-(x^2)/;-1<x≤1;
f[x_]:=f[x-2]/;x>1;
f[x_]:=f[x+2]/;x≤-1;
Plot[f[x],{x,-5,5}];
Trace[f[3]];

Plot[Piecewise[{{x^2,x<0},{x,x>0}}],{x,-2,2}];

ClearAll[f];
f[x_]:=Module[{ksi},

ksi

];
f[5]

ClearAll[myInt];
myInt[vyr_,{x1_,x2_}]:=Module[{rce,y,poc},
rce=y'[x]==vyr;
poc=y[x1]==0;
y[x2]/.NDSolve[{rce,poc},y,{x,x1,x2}][[1]]
]
myInt[(x^2),{0,1}]

Select[{1,2,4,7,6,2},#>2&]

ClearAll[test];
test[x_]:=x>2;
Select[{1,2,4,7,6,2},test]

((#)^2)&[2]

((#1)^2)+((#2)^3)&[E,Pi]

lide={ty,ona,on,soused};
Map[ostrihej,lide]
ostrihej/@lide

Apply[natocPecko,lide]

natocPecko@@lide

(∑_(i=1)^100)i

Plus@@Range[100]

Plus@@(((#)^3)&/@Range[100])

ClearAll[f];
f[x_List]:=Plus@@x;
f[x_Integer]:=(x^2);
f[x_Real]:=0.5x;
f[{a,b,c}]
f[5]
f[5.]
f[1+I]

NumericQ[a]
NumericQ[5.+I]

ClearAll[f];
f[x_?NumericQ]:=(x^2);
f[5]
f[a]

ClearAll[f];
f[x_?(#>3&)]:=(x^2);
f[5]
f[1]

0==(∑_(i=0)^1)(Subscript[a, i])*(x^i)

rce=Plus@@((Subscript[a, #])*(x^(#))&/@Range[0,5])==0
Solve[rce,x]

t0=0;
tmax=10;
n=1000;
Δt=N[((tmax-t0)/n)];
y0=5;
ClearAll[krok,derivaceY];
derivaceY[{t_,y_}]:=-y;
krok[{t_,y_}]:={t+Δt,y+Δt*derivaceY[{t,y}]};
startBod={t0,y0};

res=NestList[krok,startBod,n];
pl1=ListPlot[res,PlotRange->All,AxesLabel->{"t","y"},PlotStyle->PointSize[0.015]];
pl2=Plot[y0*(E^(-(t-t0))),{t,t0,tmax},PlotStyle->Red,PlotRange->All];
Show[pl1,pl2]

AbsoluteTime[]

Series[y[t],{t,0,5}]/.t->Δt

D[f[t,y[t]],t]

(*zde 3. prednes*)

ClearAll["Global`*"];

dosR=ToExpression["R"<>ToString[#]]->100&/@Range[4]

rceSoucastek={u3==iR4*R4,u3==R3*iR3,u1==R1*iR1,uR2==u3-u2,uR2==R2*iR2,uz==u1-u2};
rceUzly={0==iR2+iR3+iR4+iz,iR2+iz+iU==0,0==iR1+iU};
rceDohr=Union[rceSoucastek,rceUzly];
Length[rceDohr];
Solve[rceDohr,{u1,u2,u3,uR2,iR1,iR2,iR3,iR4,iU}][[1]]/.Union[dosR,{uz->10.,iz->0.1}]

rceSoucastek={u3==R3*iR3,u3-u2==R2*iR2,u2-u1==R1*iR1,uz==u3-u1};
rceUzly={iR1+iz==iuz,iR2==iR1,iuz==iR2+iR3+iOut};
rceDohr=Union[rceSoucastek,rceUzly]/.Union[dosR,{uz->10.,iz->0.1}];
u3/.Solve[rceDohr,{iR3,iR2,iR1,u3,u2,u1,iuz}][[1]]

ClearAll["Global`*"];
uz=Tanh[100*Sin[100Pi*t]];
L=0.01;
c=5*(10^(-6));
R1=1000;
R3=(10^4);
R2=500;
rceR1={u1[t]-u2[t]==R1*iR1[t]};
rceR2={u2[t]-u3[t]==R2*iR2[t]};
rceR3={u3[t]==R3*iR3[t]};
kond={iC[t]==c*u3'[t],u3[0]==0};
civka={u3[t]-u2[t]==L*iL'[t],iL[0]==0};
zdrojU={uz==u1[t]};
soucastky=Union[rceR1,rceR2,rceR3,kond,civka,zdrojU];
uzle={iR1[t]+iL[t]==iR2[t],iR2[t]==iC[t]+iL[t]+iR3[t]};
dohr=Union[soucastky,uzle];
nezname=Union[Cases[dohr,_Symbol[t],{0,∞}]]

tmax=0.1;
res=NDSolve[dohr,nezname,{t,0,tmax},StartingStepSize->(10^(-13))][[1]];
Plot[{u3[t]/.res,uz},{t,0,tmax},PlotRange->All]