Problem 20 - Advanced Engineering Mathematics

Question

Solar Collector The differential equation $$ \frac{d y}{d x}=\frac{-x+\sqrt{x^{2}+y^{2}}}{y} $$ describes the shape of a plane curve \(C\) that will reflect all incoming light beams to the same point and could be a model for the mirror of a reflecting telescope, a satellite antenna, or a solar collector. See Problem 29 in Exercises 1.3. There are several ways of solving this DE. (a) Verify that the differential equation is homogeneous (see Section 2.5). Show that the substitution \(y=u x\) yields $$ \frac{u d u}{\sqrt{1+u^{2}}\left(1-\sqrt{1+u^{2}}\right)}=\frac{d x}{x} $$ Use a CAS (or another judicious substitution) to integrate the left-hand side of the equation. Show that the curve \(C\) must be a parabola with focus at the origin and is symmetric with respect to the \(x\)-axis. (b) Show that the first differential equation can also be solved by means of the substitution \(u=x^{2}+y^{2}\).

Step-by-Step Solution

Answer

Use substitution to simplify and solve the equation, revealing its parabolic properties.

Step 1: Recognize the Equation's Homogeneity

To determine if the differential equation \( \frac{d y}{d x}=\frac{-x+\sqrt{x^{2}+y^{2}}}{y} \) is homogeneous, recognize that a differential equation is homogeneous if it can be expressed in the form of \( \frac{dy}{dx} = F\left(\frac{y}{x}\right) \). This implies that replacing \( y \) with \( vx \) gives us \( \frac{dy}{dx} = \frac{d(vx)}{dx} = v + x \frac{dv}{dx} \). This confirms that the given equation can be approached via a substitution dependent on \( \frac{y}{x} = v \).

Step 2: Apply the Substitution \( y = ux \)

By substituting \( y = ux \), we find that \( \frac{dy}{dx} = u + x\frac{du}{dx} \). Substitute \( y = ux \) into the original differential equation to obtain \( \frac{d(y)}{dx} = \frac{-x+\sqrt{x^2+(ux)^2}}{ux} = \frac{-x + x\sqrt{1+u^2}}{ux} = \frac{-1 + \sqrt{1+u^2}}{u} \). Set \( u + x\frac{du}{dx} = \frac{-1 + \sqrt{1+u^2}}{u} \).

Step 3: Simplifying Separated Variables

Rearrange the equation to get \( u du \) on one side and the other terms involving \( x \) on the other, giving \( u du = \frac{dx}{x}\cdot \frac{u}{\sqrt{1+u^2}(-1+\sqrt{1+u^2})} \).

Step 4: Integrate Both Sides

Integrate both sides: \( \int u du = \int \frac{dx}{x \sqrt{1+u^2}(-1+\sqrt{1+u^2})} \). The left integral is standard and yields \( \frac{1}{2} u^2 \). The right-hand side must be integrated using a CAS or an appropriate method.

Step 5: Apply Parabolic Characteristics

Solving the resulting equation from integration will yield a solution that reveals parabolic characteristics. Simplify to show that the equation takes the form \( y^2 = 4ax \), indicating a parabola.

Step 6: Second Substitution \( u = x^2 + y^2 \)

For part (b), use the substitution \( u = x^2 + y^2 \) which transforms the differential equation into a solvable form, potentially with separable variables for easier integration.

Key Concepts

Homogeneous Equation

A homogeneous differential equation is one where you can express the function as a ratio of two homogenous functions of the same degree. Essentially, an equation is homogeneous if it can be rewritten such that both sides possess the same degree of units. In simpler terms, if replacing \( y \) with \( vx \) (where \( v \) is a function of \( x \)) transforms the differential equation into one involving just \( v \), then it's homogeneous.
Much like classifying different gear sizes by the same diameter, we scale variables to test homogeneity. Given an equation \( \frac{dy}{dx} = F\left(\frac{y}{x}\right) \), it's homogeneous if substituting \( y = vx \) makes it solvable by reducing to \( \frac{dy}{dx} = v + x\frac{dv}{dx} \). Thus, this transformation allows us to use substitution methods to solve such equations efficiently.

Separation of Variables

Separation of Variables is a classical method, primarily used when dealing with ordinary differential equations. It facilitates solving an equation by separating the variables onto different sides of the equation. Imagine solving a puzzle by grouping pieces of the same color before assembling.
In its simplest form, if our equation is \( \frac{dy}{dx} = g(y)h(x) \), we can rearrange it to \( \frac{dy}{g(y)} = h(x)dx \). This arrangement allows us to integrate each side separately, leading to solutions of the form \( G(y) = H(x) + C \), where \( G(y) \) and \( H(x) \) are antiderivatives of \( g(y) \) and \( h(x) \), respectively.
The power of this method lies in its ability to simplify complex problems into integrable forms, thus providing direct paths to solutions.

Substitution Method

The Substitution Method is a strategic maneuver in mathematics, akin to using a decoder ring to translate puzzling messages into understandable language. This method involves replacing a complex variable with a simpler one to make an equation easier to solve.
For instance, in our given problem, the substitution \( y = ux \) was used to transform the original differential equation. This substitution simplifies the complex expression into a form more amenable to solution. It converts variables \( x \) and \( y \) into \( u \) and resolves them analytically.
This technique is especially valuable in transforming complicated differential equations involving multiple variables into single variable contexts, where traditional methods can be more easily applied. Each successful substitution is like finding the right key for a lock, opening up pathways to solutions.

Parabolic Curve

In mathematics, a parabolic curve has a unique and fascinating property – it’s shaped such that each point is equidistant from a fixed point, called the focus, and a straight line, the directrix. Parabolas have ubiquitous applications, appearing in the paths of objects under constant force, such as projectiles.
Our differential equation describes such a curve because, through mathematical transformations and simplifications, it mimics the standard equation of a parabola \( y^2 = 4ax \). This reveals the equation forms the geometric shape we know as a parabola.
The parabolic nature is not a mere trait of mathematical curiosity but a design principle leveraged in technology. Devices like satellite dishes and telescopes use parabolic shapes because they direct all incoming waves to a single focal point, optimizing reception.

Integrating Factor

An integrating factor is a potent technique in solving linear differential equations, particularly when traditional methods falter. Think of it as a multiplier that makes unsolvable equations solvable.
Consider the equation \( dy/dx + Py = Q \). Here, if you can determine a function \( \mu(x) \) such that multiplying the entire equation by \( \mu(x) \) transforms the left side into an exact derivative, the equation simplifies to \( \frac{d}{dx}[\mu(x)y] = \mu(x)Q \).
Once multiplied, it integrates seamlessly on both sides, yielding a solution. This clever twist enables us to collapse complexity into simplicity, allowing us to proceed directly to integration and find solutions that might otherwise seem elusive.