Question 1
Then find
.
Answer to Question 1:
.
See video (in two parts) for solution. Total run time about 9 minutes 30 seconds.
Part 1.
Part 2.
The following Figure shows the solution of the IVP from Question 1 plotted in (x, y) space.
The following Maple commands will solve the IVP from Question 1 and create the above Figure.
You can copy and paste these commands into Maple.
with(plots): a := -2: b := 4: c := -4: d := -2: x0 := 2: y0 := 8: tq := 0.5: dex := diff(x(t), t) = a*x(t) + b*y(t); dey := diff(y(t), t) = c*x(t) + d*y(t); de := diff(x(t), t) = a*x(t) + b*y(t), diff(y(t), t) = c*x(t) + d*y(t): de := dex, dey: ics := x(0) = x0, y(0) = y0; ans := dsolve([de, ics]): ics; ans[1]; ans[2]; evalf(subs(t = tq, ans[1])); evalf(subs(t = tq, ans[2])); xtq := rhs(evalf(subs(t = tq, ans[1]))): ytq := rhs(evalf(subs(t = tq, ans[2]))): xtq6dp := evalf(xtq, 7): ytq6dp := evalf(ytq, 7): PlotXY := plot([rhs(ans[1]), rhs(ans[2]), t = 0 .. 5], size = [0.7, 0.5], thickness = 8, gridlines, color = "brown", titlefont = ["TIMES", 24], title = typeset(dex, ", x(0) = ", x0, " \n", dey, ", y(0) = ", y0), captionfont = ["TIMES", 24], caption = typeset(ans[1], "\n", ans[2]), legend = [typeset(" ( x(t), y(t) )")], legendstyle = [font = ["TIMES", 24], location = bottom]): PlotPt := plot([xtq], [ytq], style = point, symbolsize = 12, symbol = solidcircle, color = black, legend = [typeset(" ( x(", tq, "), y(", tq, ") ) ")], legendstyle = [font = ["TIMES", 24], location = bottom]): plotText := textplot([xtq, ytq, typeset("(", xtq6dp, ", ", ytq6dp, " )")], align = {'above', 'left'}, font = ["TIMES", 24]): display({PlotPt, PlotXY, plotText});