1+1

(Sqrt[5])

(Sqrt[5.])

N[(Sqrt[5])]

N[(Sqrt[5]),100]

(a+b)*(a-b)
Expand[%]

%

c=3;
d=4;
e=c+d;

husa11=11;
11husa+husa

kyncl je vul

r*h*f

?Sin

LaplaceTransform

x=10;
ClearAll[x];
Integrate[(x^10)*Sin[(1/x)],x]

N[Pi]
Pi//N
N@Pi

120!//N

list={1,2,3,4};
Sin[list]

grafik=Plot[(t^3)Sin[5t+0.4],{t,0,10}];

grafik

DSolve[(x^2)*y''[x]+x*y'[x]+y[x]==0,y[x],x]

ClearAll[y]
DSolve[{(x^2)*y''[x]+x*y'[x]+y[x]==0,y'[1]==1,y[1]==-1},y[x],x]
Plot[Evaluate[y[x]/.%], {x,-10,10}]

Plot[-Cos[Log[x]]+Sin[Log[x]],{x,1,10}]

(*zde druhe cviceni*)
Quiet@Remove["Global`*"]

D[Tanh[x],x]
Tanh'[x]

Sin'

Plot[{D[Tanh[x],x],(1/(1+(x^2)))},{x,-5,5},AxesLabel->{"x","fce"},GridLines->Automatic,PlotStyle->{{Black,Thick},{Red,Thick}}]

Plot[{Tanh'[x],(1/(1+(x^2)))},{x,-5,5},AxesLabel->{"x","fce"},GridLines->Automatic,PlotStyle->{{Black,Thick},{Red,Thick}}]

Manipulate[
Plot[{Tanh'[k*x],(1/(1+(x^2)))},{x,-5,5},AxesLabel->{"x","fce"},GridLines->Automatic,PlotStyle->{{Black,Thick},{Red,Thick}}]
,{{k,1},0,2}]

x=5
psi=1

ClearAll[f,g,x,psi];
f[x_]=3*x*Cos[3x];
g[psi_]=((Sin[psi])/(1+(psi^2)));
Plot[g[3*f[(Sqrt[t])+1]],{t,1,10},AxesLabel->{"t","g[3*f[(Sqrt[t])+1]]]"},GridLines->Automatic,PlotStyle->{Magenta,Thick,Dashed}]

ClearAll[f,g];
f[x_]:=3*x*Cos[3x];
g[psi_]:=((Sin[psi])/(1+(psi^2)));
Plot[g[3*f[(Sqrt[t])+1]],{t,1,10},AxesLabel->{"t","g[3*f[(Sqrt[t])+1]]"},GridLines->Automatic,PlotStyle->{Magenta,Thick,Dashed}]

a==b
FullForm[%]

rovnice=(x^3)+(1/(1+(t^2)))*(x^2)+t*x-1==0;
dosadZax123=Solve[rovnice,x];
x1=x/.dosadZax123[[1]];
x2=x/.dosadZax123[[2]];
x3=x/.dosadZax123[[3]];
proPlot=Abs[x1]+Abs[x2]+Abs[x3];
Plot[proPlot,{t,1,10},AxesLabel->{"t","Abs[x1]+Abs[x2]+Abs[x3]"},GridLines->Automatic,PlotStyle->{Blue,Thick}]

x/.Solve[(x^3)==8]

Abs[{a,b,c}]

Plus@@Abs[x/.Solve[(x^3)+(1/(1+(t^2)))*(x^2)+t*x-1==0,x]]//Plot[#,{t,1,10},AxesLabel->{"t","Abs[x1]+Abs[x2]+Abs[x3]"},GridLines->Automatic,PlotStyle->{Blue,Thick}]&

ClearAll[f];
rce=(1+(t^2))*(x^4)+(x/(1+(t^4)))==Sin[1+(t^3)];
dosadZax1234=Solve[rce,x];
x1=x/.dosadZax1234[[1]];
x2=x/.dosadZax1234[[2]];
x3=x/.dosadZax1234[[3]];
x4=x/.dosadZax1234[[4]];
f[t_]=((Abs[x1]-Abs[x4])/(1+Abs[x2]+Abs[x3]));
proPlot=f[3*Cos[1+(x^2)]];
Plot[proPlot,{x,0,5},AxesLabel->{"t","((Abs[x1] - Abs[x4])/(1 + Abs[x2] + Abs[x3]))"},GridLines->Automatic,PlotStyle->{Red,Thick}]

ClearAll[f];
rce=(1+(t^2))*(x^4)+(x/(1+(t^4)))==Sin[1+(t^3)];
dosadZax1234=Solve[rce,x];
x1=x/.dosadZax1234[[1]];
x2=x/.dosadZax1234[[2]];
x3=x/.dosadZax1234[[3]];
x4=x/.dosadZax1234[[4]];
f[t_]:=((Abs[x1]-Abs[x4])/(1+Abs[x2]+Abs[x3]));

proPlot=f[3*Cos[1+(x^2)]];
Plot[proPlot,{x,0,5},AxesLabel->{"t","((Abs[x1] - Abs[x4])/(1 + Abs[x2] + Abs[x3]))"},GridLines->Automatic,PlotStyle->{Red,Thick}]

?f

(*zde treti cviceni*)
Quiet@Remove["Global`*"]

ClearAll[f]
f[a_]:=x/.FindRoot[x==Cos[a*x],{x,1}]
Plot[f[(Sqrt[t])],{t,0.25,1},PlotStyle->{Orange,Thick},AxesLabel->{"t","f((Sqrt[t]))"},GridLines->Automatic]

x/.{x->0.78569}

FindRoot

ClearAll[f]
f[a_]:=x/.FindRoot[Cos[a*x]==x,{x,0}]

x/.{x->0.9003672225897477`}

ClearAll[f]
f[a_]:=FindRoot[Cos[a*x]==x,{x,0}][[1,2]]
f[0.5]

ClearAll[f];
f[a_,b_]:=(((x^2)-(y^2))/(1+(x^2)+(y^2)))/.Solve[{a*x+((a^2)-(b^2))*y==2a,b*x+((a^2)+(b^2))*y==2b},{x,y}][[1]];
Plot[f[(Sqrt[t]),1+(Sqrt[t])],{t,0.25,1},PlotStyle->{Orange,Thick},AxesLabel->{"t","f((Sqrt[t]),1+(Sqrt[t]))"},GridLines->Automatic]

f[a_,b_]:=(((x^2)-(y^2))/(1+(x^2)+(y^2)))/.Solve[{a*x+((a^2)-(b^2))*y==2a,b*x+((a^2)+(b^2))*y==2b},{x,y}][[1]]
f[1,2]

Solve[{a*x+((a^2)-(b^2))*y==2a,b*x+((a^2)+(b^2))*y==2b},{x,y}]

x/.{x->4}
(x^2)/.{x->4}

pravidlo={x->4}
x/.pravidlo

ClearAll[f]
f[a_]:=(((x^2)+1)/((y^2)+1))/.Solve[{x+2y==a,x-7y==(a^2)},{x,y}][[1]]
Plot[f[a],{a,0,1},PlotStyle->{Orange,Thick},AxesLabel->{"a","f(a)"},GridLines->Automatic]

ClearAll[xv,yv]
{xv[t_],yv[t_]}={x[t],y[t]}/.DSolve[{x'[t]==-x[t]+y[t],y'[t]==-y[t]+x[t],x[0]==100,y[0]==-100},{x[t],y[t]},t][[1]];
xv[1]
yv[1]

x[1]/.res

res

xv[1]

ClearAll[xv,yv]
res=DSolve[{x'[t]==-x[t]+y[t],y'[t]==-y[t]+x[t],x[0]==100,y[0]==-100},{x[t],y[t]},t][[1]];
xv[t_]=x[t]/.res;
yv[t_]=y[t]/.res;
xv[1]
yv[1]

res

Function[{t},100 (E^(-2 t))][1]

ClearAll[xv,yv]
res=DSolve[{x'[t]==-x[t]+y[t],y'[t]==-y[t]+x[t],x[0]==100,y[0]==-100},{x,y},t][[1]];
xv[t_]:=x[t]/.res;
yv[t_]:=y[t]/.res;
xv[1]
yv[1]

b=Sin[x];
ClearAll[f];
f[x_]:=b;
f[1]

?f

b=Sin[x];
ClearAll[f];
f[x_]=b;
f[1]

?f

b=Sin[x];
ClearAll[f];
f[xx_]:=b/.x->xx;
f[1]

Quiet@Remove[Evaluate[Context[]<>"*"]]

R1=100;
R2=200;
iz=0.01*Sin[314t+0.24];
uz=5*Sin[314t-.569];
L=0.05;
c=15*(10^(-6));
rce={iu[t]==iz+iR1[t],
iz==ic[t]+iL[t],
ic[t]==iu[t]+iR2[t],
u1[t]==R1*iR1[t],
u2[t]==R2*iR2[t],
uC[t]==u3[t]-u2[t],
ic[t]==c*uC'[t],
uz==u1[t]-u2[t],
u3[t]==L*iL'[t]
};
nezname=Union[Cases[rce,_Symbol[t],{0,∞}]];
pocs=Union[Cases[rce,a_'[t]:>a[0]==0,{0,∞}]];
rceDohr=Union[rce,pocs];
res=NDSolve[rceDohr,nezname,{t,0,0.08}][[1]];
Plot[ic[t]/.res,{t,0,0.08}]

pocs=Cases[rce,a_'[t]:>a[0]==0,{0,∞}];

pocs=Cases[rce,a_'[t],{0,∞}]

list={1,5.,x,(y^z),Sin[5],ic[t],uc'[t]}
Cases[list,_Symbol[t],{0,∞}]

TreeForm[list]