# A system of differential equations [closed]

Consider the following system of differential equations: $$x'(t) = a\, x(t) - b \,x(t) y(t),$$ and $$y'(t) = c\, x(t) y(t) - d \,y(t),$$ with $$x(0) = x_0, y(0) = y_0$$

For example, we may have:

a = 2.2; b = 0.03; c = 1.4; d = 0.02;
eq1 = x'[t] == a*x[t] - b*x[t]*y[t]
eq2 = y'[t] == -c*y[t] + d*y[t]*x[t]
sol = NDSolve[{eq1, eq2, x[0] == 150, y[0] == 10}, {x, y}, {t, 0, 30}]
gr1 = Plot[Evaluate[{x[t], y[t]} /. sol], {t, 0, 30}, PlotStyle -> {{Thick,
Hue[0.7]}, { Dashing[{0.03}], Hue[1]}}, Ticks -> {{{0.01, 0}, {10, "time"},
20}, {500, {700, "number"}, 1000}}, LabelStyle -> {FontFamily -> "Times",
FontSize -> 16}, PlotLabel -> "solid blue: rabbits, dashed red: foxes"];
gr2 = ParametricPlot[Evaluate[{x[t], y[t]} /. sol], {t, 0, 30}, PlotStyle ->
{Thick, Hue[0.3]}, Ticks -> {{{0.01, 0}, 500, {700, "rabbits"}, 1000}, {500,
{700, "foxes"}, 1000}}, LabelStyle -> {FontFamily -> "Times", FontSize ->
16}];
sp = StreamPlot[{2.2 x - 0.03 y x, -1.4 y + 0.02 y x}, {x, 0, 200}, {y, 0,
150}, StreamScale -> Large]
GraphicsRow[{gr1, gr2, sp}]
f[x_, y_] = {2.2 x - 0.03 x y, -1.4 y + 0.02 x y};
Show[VectorPlot[f[x, y], {x, 0, 4}, {y, 0, 4}], Frame -> True,
BaseStyle -> {FontFamily -> "Times", FontSize -> 14}]
Show[VectorPlot[
f[x, y]/(10^-8 + Norm[f[x, y]]), {x, 0, 4}, {y, 0, 4}],
Frame -> True, BaseStyle -> {FontFamily -> "Times", FontSize -> 14}]


I can't use a numerical method (e.g. Euler method) instead of NDSolve. I need the same method for $$m x''(t)=-g x(t)^3$$ Is there any idea?

• What do you mean by "I can't use a numerical method (e.g. Euler method) instead of NDSolve. I need the same method for $mx′′(t)=−gx(t)^3$"? What exactly do you need? Commented Jan 19, 2020 at 10:06
• I don't need to use NDSolve for solving the above system, that's all Commented Jan 19, 2020 at 10:07
• Do you mean you need to solve the problem without NDSolve? Commented Jan 19, 2020 at 10:13
• yes, without NDSolve Commented Jan 19, 2020 at 10:15
• But you've already asked a question about Euler method and received an answer here. Solving a ODE and a system of ODE with Euler method are just the same. If you have difficulty in understanding that answer, you should continue asking in the comment under that answer, rather than ask essentially the same question again. Commented Jan 19, 2020 at 11:06

DSolve[{y'[x] == (-c*y[x] + d*y[x]*x)/(a*x - b*x*y[x])}, y, x]
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