A family of curves is obtained from the cubic parabola y = x^3/8 by translations along the x-axis. Find an equation of the orthogonal family. Solve the problem along the following lines. (a). Write down the equation of the given family. It should contain an arbitrary constant. (b). Use Desmos to draw several curves from the family (4-6 curves). Provide a screenshot of the graph and a command line. (c). Find a differential equation describing the curves of the family. (d). Compose a differential equation for the orthogonal family. (e). Solve the differential equation for the orthogonal family and establish the equation for the orthogonal family in explicit form (as a function y = y(x)). (f) Draw several curves from the orthogonal family on the same picture where the original family was drawn. Provide a screenshot of the graph and a command line. Are the curves from two families orthogonal indeed at points of intersections? Make comments on your picture.
A family of curves is obtained from the cubic parabola y = x^3/8 by
translations along the x-axis. Find an equation of the orthogonal family. Solve the problem along the following lines.
(a). Write down the equation of the given family. It should contain an arbitrary constant. (b). Use Desmos to draw several curves from the family (4-6 curves). Provide a screenshot
of the graph and a command line.
(c). Find a differential equation describing the curves of the family.
(d). Compose a differential equation for the orthogonal family.
(e). Solve the differential equation for the orthogonal family and establish the equation for the orthogonal family in explicit form (as a function y = y(x)).
(f) Draw several curves from the orthogonal family on the same picture where the original family was drawn. Provide a screenshot of the graph and a command line. Are the curves from two families orthogonal indeed at points of intersections? Make comments on your picture.
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