Orthogonal Families of Curves — First-Order ODE

Two families of curves in the plane are called orthogonal if every curve of one family meets every curve of the other family at a right angle.

Usually, such a family is the general solution of a first-order ODE of the form

\( y' = f(x,y). \)

Two curves meet at a right angle if their slopes (the values of their derivatives) at the intersection point satisfy

\( k_1 k_2 = -1, \)

that is, the product of the slopes is \( -1 \). Therefore, for a curve of the orthogonal family, the slope at any point must be

\( y' = -\frac{1}{f(x,y)}. \)

This differential equation defines the curves orthogonal to those of the original family.

Two orthogonal families of lines: \( y - x = C \) and \( y + x = C \) — the slopes \( 1 \) and \( -1 \) multiply to \( -1 \)

Finding the orthogonal family by solving an ODE

The circles \( x^2+y^2=C \) and the rays through the origin form orthogonal families

For a family of curves that solves the ODE \( y' = f(x,y) \), the orthogonal family satisfies \( y' = -\dfrac{1}{f(x,y)} \), wherever the expression is defined. In practice:

1. Differentiate the family equation \( f(x,y,c) = 0 \) to obtain an equation involving \( y' \).

2. If the new equation still contains the constant \( c \), eliminate it by isolating \( c \) from the original equation.

3. Isolate \( y' = g(x,y) \).

4. The orthogonal family must satisfy \( y_\perp' = -\dfrac{1}{y'} \), but at the same point \( (x,y) \) — so the new equation for \( y_\perp \) is

\( y_\perp' = \frac{-1}{g(x,\, y_\perp)}. \)

Exercise

Find the family of curves orthogonal to the parabolic family \( y^2 = Cx \).

Solution. Differentiating both sides gives

\( 2y\, y' = C. \)

Isolate the constant: \( C = \dfrac{y^2}{x} \), and substitute it into the derivative equation:

\( 2y\, y' = \frac{y^2}{x} \;\Rightarrow\; y' = \frac{y}{2x}. \)

This is the differential equation of the parabolic family. The orthogonal equation is therefore

\( y' = -\frac{2x}{y}. \)

Solving it: \( y\, y' = -2x \), so

\( \frac{1}{2} y^2 = -x^2 + C \;\Rightarrow\; x^2 + \frac{y^2}{2} = C, \)

which is a family of ellipses.

The parabolas \( y^2 = Cx \) and their orthogonal family — the ellipses \( x^2 + \frac{y^2}{2} = C \)