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Determine ODE with Slope Field: Quadratic
dgl-09-03
multiple
195
randRangeExclude(-8,8,[0]) randRangeExclude(-8,8,[0,-A,A]) A+B A*B

The direction field of the ODE y' = y^2 + {\color{orange}a}y + {\color{teal}b} is given below.

Determine \color{orange}a and \color{teal}b .

style({ stroke: "black", strokeWidth: 0.05 }); graphInit({ range: [[-8, 8], [-10,10]], scale: [22, 22], axisArrows: "->", tickStep: 1, }); label([0,A], "\\color{red}A", "left"); label([0,B], "\\color{blue}B", "left"); // Vektorfeld for(var i = -8; i <= 8; i+=1) { for(var j = -9; j <= 9; j+=1) { var dy = (j - A)*( j - B)/15; line([i - 0.25, j - dy*0.25], [i + 0.25, j + dy*0.25], { arrows: "", strokeWidth: 1, stroke: "gray" }); } }

x \color{orange}a = -T
y \color{teal}b = D

At a given point (x_0,y_0), the slope field shows a tiny piece of the tangent of a solution of the ODE, on which (x_0,y_0) lies.

The slope of the tangent is determined by the value of the right-hand side of the ODE at the point (x_0,y_0).

At the points \color{red}A und \color{blue}B the tangent slopes are zero.

Thus \color{red}A and \color{blue}B must be the roots of the right-hand side y' = y^2 + {\color{orange}a}y + {\color{teal}b}.

With (y- A) (y- B) = y^2 +{\color{orange}a}y + {\color{teal}b} one gets {\color{orange}a} = -T and {\color{teal}b} =D.