Question

Find the orthogonal trajectories of each of the following families of curves. In each case sketch several of the given curves and several of their orthogonal trajectories. Be careful to eliminate the constant from \(d y / d x\) for the original curves; this constant takes different values for different curves of the original family and you want an expression for \(d y / d x\) which is valid for all curves of the family crossed by the orthogonal trajectory you are trying to find. \(y=k x^{n}\). (Assume that \(n\) is a given number; the different curves of the family have, different values of \(k_{2}\) )

Short answer

The orthogonal trajectories are given by \( n y^2 + x^2 = D \).

Step-by-step solution

Differentiate the Original Curve

Given the curve equation is: \( y = k x^n \) Differentiate this equation with respect to \(x\) to find \(\frac{dy}{dx}\). Using the chain rule of differentiation, \( \frac{dy}{dx} = k n x^{n-1} \)

Eliminate the Constant

Since \(k\) is constant and varies for the given family of curves, eliminate \(k\) from the derivative. Solve the original equation for \(k\): \( k = \frac{y}{x^n} \). Substitute this result into \( \frac{dy}{dx} \):\( \frac{dy}{dx} = \left(\frac{y}{x^n}\right) n x^{n-1} = \frac{n y}{x} \)

Find the Slope of the Orthogonal Trajectories

The orthogonal trajectory must satisfy the condition that its slope \( \frac{dy'}{dx}\) is the negative reciprocal of the slope of the original curve. So,\( \frac{dy'}{dx} = -\frac{x}{n y} \)

Separate the Variables

Separate the variables in \(\frac{dy'}{dx}\) to solve for the orthogonal trajectories: \( n y dy' = -x dx \)

Integrate Both Sides

Integrate both sides to find the general solution: \( \int n y \, dy' = -\int x \, dx \)This gives: \( \frac{n y^2}{2} = -\frac{x^2}{2} + C \) where \(C\) is the constant of integration.

Simplify the Equation

Rearrange and simplify the equation: \( n y^2 + x^2 = D \) where \( D \) is a new constant.

Conclusion: Orthogonal Trajectories

The orthogonal trajectories for the family of curves \( y = k x^n \) are given by the equation: \( n y^2 + x^2 = D \)

Key concepts

calculus

Calculus is a branch of mathematics that deals with continuous change. It's divided into two main parts: differential calculus and integral calculus. In this exercise, we're focusing on differential calculus, where we use differentiation to find the rate at which a function is changing at any given point. This is crucial for understanding the slopes of curves, especially when finding orthogonal trajectories. By differentiating the function, we obtain the slope of the given curves which is essential in solving our problem. Differentiation involves applying rules like the chain rule to break down complex expressions into their simplest form.

differential equations

Curve sketching allows us to visualize the behavior of functions and their interactions on a graph. In our problem, we’re dealing with the family of curves given by \( y = k x^n \) and the orthogonal trajectories that intersect them at right angles. Sketching multiple curves from the family will show different values of \( k \). By also sketching their orthogonal trajectories, we can see the perpendicular intersections. This graphical representation helps in comprehending how the curves and their trajectories relate to each other.