(*č. 1*)

a1 = VectorPlot[{2x-y,4x-3y},{x,-5,5},{y,-5,5}]

b1=Plot[{y = x,y = 4x},{x,-5,5}, {PlotStyle-> {Red}}] ;(* červená = směr komponenty e^x*)

Show[a1,b1, ParametricPlot[{ Exp[t] + Exp[-2t],Exp[t]+4Exp[-2t]},{t,-1,2}, {PlotStyle-> {Green}}], ParametricPlot[{ Exp[t] - Exp[-2t],Exp[t]-4Exp[-2t]},{t,-1,2}, {PlotStyle-> {Blue}}]]

(*č. 2*)

a2= VectorPlot[{x+y,-2x+3y},{x,-2,2},{y,-2,2}]

Manipulate[Show[a2,ParametricPlot[{a  Exp[2t] Sin[t],a Exp[2t]( Cos[t]+Sin[t])},{t,-1,1}, {PlotStyle-> {Green}}]], {a,-2,2}] (*reseni pouzivajici jen prvni slozku reseni*)

(*řešení soustavy přes maticovou exponenciálu:*)
a = {{2,-1},{4,-3}}
{l,v}={Eigenvalues[a],Eigenvectors[a]}
d = DiagonalMatrix[Exp[t l]]
p = Transpose[v]
 p.d.Inverse[p].{y1[0],y2[0]}//MatrixForm (*{y1[0],y2[0]} = poč. podm*)
(({{(-(1/3) (E^(-2 t))+((4 (E^t))/3)) y1[0]+(((E^(-2 t))/3)-((E^t)/3)) y2[0]}, {(-(4/3) (E^(-2 t))+((4 (E^t))/3)) y1[0]+(((4 (E^(-2 t)))/3)-((E^t)/3)) y2[0]}}))