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.
Video Player
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Part 2.
Video Player
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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.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 | 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}); |