25+25

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

DSolve[y''[x]+3y'[x]+y[x]==Sin[5x],y[x],x]

Series[f[x],{x,x0,5}]

ClearAll[f,x];
f[x_]=(x^2);
f[2]

ClearAll[f,x];
f[x_]=Integrate[(2^x)*Sin[x],x];
f[2]

NestList[Cos,0.,50]//ListPlot

ClearAll[EuSt];
(*resme y'[t]==-5*y[t], y[t=0]==1*)
tmax=1.;
nSteps=10;
Δt=(tmax/nSteps);
EuSt[{t_,y_}]:={t+Δt,y+(-5y)*Δt};
res=NestList[EuSt,{0,1},nSteps];
pl1=ListPlot[res,AxesLabel->{"t (s)","u (V)"},PlotStyle->{PointSize[0.015],Hue[0]}];
pl2=Plot[Exp[-5t],{t,0,tmax},PlotStyle->{Black,Thickness[0.006]}];
(*Show[pl1,pl2]*)
res2=NestList[RgSt,{0,1},nSteps];
pl3=ListPlot[res2,AxesLabel->{"t (s)","u (V)"},PlotStyle->{PointSize[0.015],Hue[0.6]}];
Show[pl1,pl2,pl3]

Manipulate[
Show[Graphics[{Hue[hue],Disk[{0,0},{3,4}]}]]
,{{hue,0.5},0,1},ControlPlacement->Top];

derUc[{t_,uc_}]:=-5uc;
RgSt[pom:{t_,uc_}]:=Module[{k1,k2,k3,k4},k1=derUc[pom];k2=derUc[{t+((Δt)/2),uc+((Δt)/2)*k1}];k3=derUc[{t+((Δt)/2),uc+((Δt)/2)*k2}];k4=derUc[{t+Δt,uc+Δt*k3}];N[pom+{Δt,((Δt)/6)*(k1+2k2+2k3+k4)}]];

res2=NestList[RgSt,{0,1},nSteps];
pl3=ListPlot[res,AxesLabel->{"t (s)","u (V)"},PlotStyle->{PointSize[0.01],Hue[0.6]}];
Show[pl1,pl2]

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

rce=y''[t]+5y'[t]+y[t]==u[t];
laplrce=LaplaceTransform[rce,t,p]/.{LaplaceTransform[y[t],t,p]->Y[p],LaplaceTransform[u[t],t,p]->U[p]}
Solve[laplrce,Y[p]]

Solve[{e==u-y,e*(1/(1+2p))*(1/(p+3))==y},y,e]
Apart[(y/u)/.%[[1]]]

Apart[(1/((p^2)+5p-6))]

Factor[(1/((p^2)+5p-6))]

ClearAll[u1];
u1[t_]:=5*Tanh[100*Sin[2*Pi*1000t]];
Plot[u1[t],{t,0,0.05},PlotPoints->300];
R1=20;
R2=1000;
c=(10^(-6));
L=0.05;
odporR1={R1*iR1[t]==u1[t]-u2[t]};
civka={u2[t]==L*iL'[t],iL[0]==0};
kond={iC[t]==c*D[u3[t]-u2[t],t],u3[0]-u2[0]==0};
uzel2={iR1[t]+iC[t]==iL[t]};
odporR2={R2*iR2[t]==u3[t]};
uzel3={0==iC[t]+iR2[t]};
rceCelkem=Union[odporR1,odporR2,civka,kond,uzel2,uzel3];
Length[rceCelkem];
nezname=Union@Cases[rceCelkem,_Symbol[t],{0,∞}];
tmax=0.01;
NDSolve[rceCelkem,nezname,{t,0,0.01}]