Zadanie 2.

calka = (∫_0^x) Sin[t^2] \[DifferentialD]t

lewa = 2/x * calka

prawa = 1 - (x/3)^2

w1 = Plot[lewa, {x, -5, 5}]

w2 = Plot[prawa, {x, -5, 5},  PlotStyle -> Red]

Show[w1, w2]

FindRoot[lewa == prawa, {x, -1}]

FindRoot[lewa == prawa, {x, 1}]

Zadanie 3.

m = 1;
V0 = 100;
k = 0.1;
g = 9.81;

rownanie1  =  
      m x''[t]  ==  - k x'[
            t]
rownanie2  =  
      m z''[
            t]  == -m g  - k z'[
              t]

α = 15;

Clear[α]

rozw = 
      Table[{α, 
          NDSolve[{rownanie1, rownanie2, 
              x[0] == 0, x'[0] == 
                V0 Sin[α], z'[0] == V0 Cos[α], 
              z[0] == 0}, {x[t], z[t]}, {t, 0, 50}]}, {α, 0, 
          Pi/2, 0.05}]

num = Dimensions[rozw][[1]]

dlax = 
      Table[x[t] /. rozw[[ii]][[2]][[1]], {ii, 1, 
          num}]

dlaz = 
      Table[z[t] /. rozw[[ii]][[2]][[1]], {ii, 1, 
          num}]

wykres = 
      ParametricPlot[
        Evaluate[Table[{dlax[[ii]], dlaz[[ii]]}, {ii, 1, num}]], {t, 
          0, 20}, AspectRatio -> 1]

zasiegTBL=Table[{rozw[[ii]][[1]], 
          dlax[[ii]] /.  
            FindRoot[dlaz[[ii]] == 0, {t, 20}]}, {ii, 1, 
          num}]

zasiegF  =  Interpolation[zasiegTBL]

FindMaximum[zasiegF[α], {α, Pi/4}]

Metoda 2

Clear[α, m, g, k, V0]

rownanie1  =  
      m x''[t]  ==  - k x'[
            t]
rownanie2  =  
      m z''[
            t]  == -m g  - k z'[
              t]

rozwanalityczne = 
      DSolve[{rownanie1, rownanie2, 
          x[0] == 0, x'[0] == 
            V0 Sin[α], z'[0] == V0 Cos[α], 
          z[0] == 0}, {x[t], z[t]}, t]

Z  =  z[t]  /.  rozwanalityczne[[1]]

X  =  x[t]  /.  rozwanalityczne[[1]]

tmax = t /. Solve[Z == 0, t][[1]]

zas = X  /.  t -> tmax  // FullSimplify

Plot[
      zas  /. {k -> 0.1, g -> 9.81, m -> 1, 
          V0 -> 100}, {α, 0, Pi/2}]

D[zas, α]  // FullSimplify

FindMaximum[
      zas  /. {k -> 0.1, g -> 9.81, m -> 1, 
          V0 -> 100}, {α, 1}]

Zadanie 1.

Clear[α]

n = {Cos[ϕ] Sin[ν], Sin[ϕ] Sin[ν], 
        Cos[ν]}

A = 
      Table[(∑_(k = 1)^3) Signature[{i, j, k}]* n[[k]],  {i, 1, 
          3}, {j, 1, 3}]

Oo = MatrixExp[α A] ;

Oo  // MatrixForm

Oo  /.  α -> 0  // Simplify  // 
      MatrixForm

Oo  /.  ν -> 0  // ComplexExpand  // Simplify  // 
      MatrixForm

Oo  /.  {ν -> Pi/2, ϕ -> 0}  // 
          ComplexExpand // Simplify  // MatrixForm

Oo  /.  {ν -> Pi/2, ϕ -> Pi/2}  // 
        Simplify  // MatrixForm

Det[Oo]  // Simplify

Tr[Oo]  // ComplexExpand  // Simplify

Eigenvalues[Oo]   // FullSimplify