Show that the differential equations y^'=3 y / x and y^'=-x / 3 y define orthogonal families of

step 1 So the octagonal family means that the slope, the product of the slot, we equal to minus 1. Here

Show that the differential equations y^'=3 y / x and y^'=-x...

Show that the differential equations $y^{\prime}=3 y / x$ and $y^{\prime}=-x / 3 y$ define orthogonal families of curves; that is, the graphs of solutions to the first equation intersect the graphs of the solutions to the second equation in right angles (Figure 10$) .$ Find these curves explicitly.

Key Concepts

Integration and Families of Curves

When solving differential equations, integration is used to accumulate the effect of the differential relationship, resulting in a general solution that includes an arbitrary constant. This constant represents a family of curves, with each member corresponding to a particular value. By integrating each separated differential equation, one obtains an equation representing a family of curves, which is essential in analyzing the overall behavior and relationships between different families, such as in the case of orthogonal trajectories.

Negative Reciprocal Relationship of Slopes

The condition for two curves to be orthogonal at the point of intersection is that the product of their slopes must equal -1. This stems from the geometric fact that the tangent lines to the curves at the point of intersection must be perpendicular. In the context of differential equations, if one family of curves has a derivative m, then the corresponding orthogonal family must have a derivative -1/m, ensuring that the angle between the tangent lines is 90 degrees.

Separable Differential Equations

A separable differential equation is one in which the variables can be rearranged so that each side of the equation contains only one variable and its differential. This method relies on rewriting the equation in the form f(y) dy = g(x) dx, which can then be integrated on both sides to obtain an implicit or explicit solution. It is a fundamental technique for solving first?order ordinary differential equations, particularly when the rate of change of the unknown function can be expressed as a product or quotient of functions of each variable separately.

Orthogonal Trajectories

Orthogonal trajectories are families of curves that intersect a given family of curves at right angles. The key idea is that if one family of curves has a slope given by a differential equation, then the slopes of the curves in the orthogonal family are the negative reciprocals of these slopes (assuming the slopes are defined and nonzero). This concept is widely used in the study of differential equations to describe curves that maintain a perpendicular relationship to another family of curves.

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