Question

Question: When all the curves in a family \(G\left( {x,y,{c_1}} \right) = 0\)intersect orthogonally all the curves in another family\(H\left( {x,y,{c_2}} \right) = 0\), the families are said to be orthogonal trajectories of each other. See Figure 3.R.5. If \(dy/dx = f(x,y)\)is the differential equation of one family, then the differential equation for the orthogonal trajectories of this family is\(dy/dx = - 1/f(x,y)\). In Problems\(\;{\bf{15}} - {\bf{18}}\)find the differential equation of the given family by computing\(dy/dx\)and eliminating \({c_1}\)from this equation. Then find the orthogonal trajectories of the family. Use a graphing utility to graph both families on the same set of coordinate axes.

\({x^2} - 2{y^2} = {c_1}\)

Short answer

\(y = {c_2}{x^{ - 2}}\)

Step-by-step solution

Definition

Orthogonal trajectory family of curves that intersect another family of curves at right angles.

Differential equation of given family

Consider \({x^2} - 2{y^2} = {c_1}\) -------(1)

If\(\frac{{dy}}{{dx}} = f(x,y)\)is the differential equation of one family, then the differential equation for the orthogonal trajectories of this family is\(\frac{{dy}}{{dx}} = - \frac{1}{{f(x,y)}}\).

Differentiate equation (1) w.r.t\(.{\rm{ x}}\).

\(\begin{array}{l}2x - 4y\frac{{dy}}{{dx}} = 0\\\frac{{dy}}{{dx}} = \frac{x}{{2y}}\end{array}\)

Differential equation of the given family is\(\frac{{dy}}{{dx}} = \frac{x}{{2y}}\).

Differential equation of orthogonal trajectory

Differential equation for the orthogonal trajectories of the given family is

\(\begin{array}{l}\frac{{dy}}{{dx}} = - \frac{1}{{f(x,y)}}\\\frac{{dy}}{{dx}} = \frac{{ - 1}}{{\left( {\frac{x}{{2y}}} \right)}}\\\frac{{dy}}{{dx}} = - \frac{{2y}}{x}\end{array}\)

Solve the above equation.

\(\begin{array}{c}\int {\frac{1}{y}} dy = \int - \frac{2}{x}dx\\\ln y = - 2\ln x + d\;\;\;(d{\rm{ is constant }})\\y = {e^{ - 2\ln x}}{e^d}\\y = {c_2}{x^{ - 2}}\left( {{\rm{ replace }}{e^d}{\rm{ by }}{c_2}} \right)\end{array}\)

The orthogonal trajectories of the given family is \(y = {c_2}{x^{ - 2}}\)