# Trajectory plot on phase plane for a desired initial conditions

I have a system of ordinary differential equations. dx/dt=-x and dy/dt=2x-2y

I am using StreamPlot to plot the phase portrait which gives me this I want to pic some initial condition and study that whether the solution leads to the equilibrium point(stable one) or not. I am trying a lot but didn't getting the required result as in the picture below I require the blue arrowed lines that are solid and pass through the equilibrium points ie (0,0) here in the pic. Thanks in advance for the help.

Only part of the work. It is hope that can help you to draw the complete picture.

Here we use ParametricNDSolve to solve the curves which pass through point $$(a,b)$$.

We select some points such as $$(1,0)$$,$$(2,0)$$,$$(3,1)$$ etc.

sols = ParametricNDSolve[{D[x[t], t] == -x[t],
D[y[t], t] == 2 x[t] - 2 y[t], x == a, y == b}, {x,
y}, {t, -10, 10}, {a, b}];
f[a_, b_][t_] := {x[a, b][t], y[a, b][t]} /. sols;
lines1 = ParametricPlot[{f[1, 0][t], f[2, 0][t], f[3, 1][t]}, {t, -.3,
10}, Epilog -> {Arrow[{f[1, 0][-.2], f[1, 0][-.1]}],
Arrow[{f[2, 0][-.2], f[2, 0][-.1]}],
Arrow[{f[3, 1][-.2], f[3, 1][-.1]}]}, PlotStyle -> Blue];
lines2 = ParametricPlot[{f[.2, 2][t], f[.3, 2][t],
f[.5, 2][t]}, {t, -.3, 10}, PlotStyle -> Red];
lines3 = ParametricPlot[{f[-1, 0][t], f[-2, 0][t],
f[-3, 0][t]}, {t, -.1, 10}, PlotStyle -> Green];
lines4 = ParametricPlot[{f[-.5, -2][t], f[-.8, -3][t],
f[-1, -3][t]}, {t, -.1, 10}, PlotStyle -> Orange];
Show[lines1, lines2, lines3, lines4, PlotRange -> All] Move the locator to change the initial conditions.

   gr = StreamPlot[{-x, 2 x - 2 y}, {x, -2, 2}, {y, -2, 2}];
fun[x_, x0_, y0_] =
y[x] /. DSolve[{y'[x] == -2 (x - y[x])/x, y[x0] == y0}, y, x];
Manipulate[
Show[gr,
Plot[fun[x, p[], p[]], {x, p[], 0},
PlotStyle -> {Red, Thickness[0.01]}],
Graphics[{Green, PointSize[0.02], Point[{0, 0}]}]]
, {{p, {2, 1}}, Locator}] I found another way that need not solved differential equation,only change the style of StreamPlot such as StreamScale and StreamPoints

(*pts=Tuples[Range[-2,2,.8],2];*)

pts = {{1, 0}, {1.5, 0}, {2, 0}, {.2, 2}, {.5, 2}, {1, 2}, {-2,
0}, {-1.5, 0}, {-1, 0}, {-.8, -2}, {-.5, -2}, {-.2, -2}};
stream = StreamPlot[{-x, 2 x - 2 y}, {x, -2.5, 2.5}, {y, -2.5, 2.5},
StreamScale -> {Full, Automatic, Automatic}, StreamPoints -> pts,
Axes -> True, Frame -> False, StreamStyle -> Blue,
StreamColorFunction -> False, AxesLabel -> {x, y}];
Show[Plot[x, {x, -1.8, 1.8}, PlotStyle -> Green,
Epilog -> {Green, Text[y == x, {2, 1.8}, Left]},
AspectRatio -> Automatic, PlotRange -> 2, • StreamScale -> {{0.5, 0}, 50, .025, Automatic} Dec 20, 2020 at 4:26
• StreamScale -> {{.4, 0}, 50, .025, # &} Dec 20, 2020 at 4:37