Krotki wstep do DSolve

Clear[x0]
sol1=DSolve[{x''[t]+a x'[t]+b x[t]==c Sin[t],x[0]==0,x'[0]==v0},x[t],t][[1,1,2]]//FullSimplify
Manipulate[ParametricPlot[{#,D[#,t]}&[%]//Evaluate,{t,0,300},PlotRange->3,AxesLabel->{"x(t)","v(t)"}],{v0,0,10},{a,0,10},{b,0,1}]

sol=NDSolve[{x''[t]+x'[t]+x[t]==Sin[t],x[0]==0,x'[0]==3},x[t],{t,0,10}][[1,1,2]]//FullSimplify
ParametricPlot[{#,D[#,t]}&[%]//Evaluate,{t,0,10},PlotRange->3,AxesLabel->{"x(t)","v(t)"}]

FullForm[sol]

D[sol,t]
Integrate[sol,t]
Plot[{%,%%},{t,0,10}]

Niespodzianka

Plot[Sin[n Pi]/n,{n,-3,3}]

Sum[Sin[n Pi]/n,{n,-3,3}]

Sum[Sin[n Pi]/n,{n,-∞,∞}]

Sum[Sin[n Pi]/n,{n,-3,∞}]

Integrate[Sin[n Pi]/n,{n,-∞,∞}]

M={{0,0,0,I,0,-I,0},{0,0,0,0,-2 I,0,2 I},{0,0,0,0,0,0,0},{2 I,0,0,0,0,0,0},{0,-I,0,0,0,0,0},{-2 I,0,0,0,0,0,0},{0,I,0,0,0,0,0}};
TraditionalForm[M]

TableForm[{#[[1]],MatrixForm[#[[2]]]}&/@Transpose@Eigensystem[{{0,0,0,I,0,-I,0},{0,0,0,0,-2 I,0,2 I},{0,0,0,0,0,0,0},{2 I,0,0,0,0,0,0},{0,-I,0,0,0,0,0},{-2 I,0,0,0,0,0,0},{0,I,0,0,0,0,0}}]]

Zadania