Problem 19 - Differential Equations and Linear Algebra

Question

\(m\) denotes a fixed nonzero constant, and \(c\) is the constant distinguishing the different curves in the given family. In each case, find the equation of the orthogonal trajectories. $$y^{2}=m x+c$$

Step-by-Step Solution

Answer

The equation of the orthogonal trajectories of the family of curves \(y^2 = mx + c\) is \(y^2 = -mx + Km\), where \(K\) is an arbitrary constant.

Step 1: Find the derivative y'

First, we need to find the derivative of the given equation with respect to \(x\). The implicit differentiation is done as follows: \[\frac{d}{dx}(y^2) = \frac{d}{dx}(mx + c) \] \[2yy' = m\] Now, let's solve for \(y'\): \[y' = \frac{m}{2y}\]

Step 2: Find the differential equation for orthogonal trajectories

Since orthogonal trajectories intersect the given curves at right angles, we should take the negative reciprocal of the slope found in Step 1. Let's denote the negative reciprocal as \(p\): \[p = -\frac{1}{y'} = -\frac{2y}{m}\] Now, we need to find the differential equation in terms of \(y\), \(x\), and \(p\): \[p = \frac{dy}{dx}\] \[-\frac{2y}{m} = \frac{dy}{dx}\]

Step 3: Solve the differential equation

We will now solve the given differential equation: \[-\frac{2y}{m} = \frac{dy}{dx}\] To solve this first-order differential equation, we can use the separation of variables method: \[-\frac{2y}{m} dy = dx\] Now we integrate both sides: \[\int -\frac{2y}{m} dy = \int dx\] \[-\frac{1}{m} y^2 = x + K\] Now, let's rearrange the equation to find the general equation of the orthogonal trajectories: \[y^2 = -mx + Km\] The equation of the orthogonal trajectories of the family of curves \(y^2 = mx + c\) is \(y^2 = -mx + Km\), where \(K\) is an arbitrary constant.

Key Concepts

Differential Equations

A differential equation is a mathematical equation that relates a function to its derivatives. It essentially expresses how a rate of change (a derivative) affects the overall behavior of a function. These equations are fundamental in math because they describe a wide range of phenomena, such as motion, growth, decay, and waves.

There are many types of differential equations, but they can generally be classified by their order, which is determined by the highest derivative present in the equation. They can also be linear or nonlinear, depending on whether the function and its derivatives appear to the first power and are not multiplied together. Solving a differential equation typically involves finding a function (or set of functions) that satisfy the equation, often using various techniques including separation of variables and integration.

Implicit Differentiation

When dealing with equations that define functions implicitly rather than explicitly, implicit differentiation becomes a necessary technique. An explicit function directly gives the output in terms of the input, like \(y = x^2+3\). However, an implicit function intertwines the dependent and independent variables, such as \(x^2 + y^2 = 1\).

In implicit differentiation, we differentiate both sides of the equation with respect to the independent variable, treating the dependent variable as a function of that independent variable. In the context of orthogonal trajectories, which often come in the implicit form, this differentiation method allows us to find the slope of the tangent line to the curve at any given point.

Separation of Variables

The method of separation of variables is a common technique for solving first-order differential equations. The process involves rearranging the equation so that each variable appears on a different side of the equation. This maneuver allows us to integrate each side with respect to its own variable, leading to a solution that includes an arbitrary constant.

For the example problem, separation of variables allows us to solve an equation that describes the orthogonal trajectories to the family of curves given. This is because we can separate the variables and then integrate to find a relationship between \(x\) and \(y\) that satisfies the differential equation of the orthogonal trajectories.

First-Order Differential Equation

A first-order differential equation involves only the first derivative of the function and no higher derivatives. The equation can often be written in the form \(\frac{dy}{dx} = f(x, y)\), where \(f(x, y)\) can be any expression involving \(x\) and \(y\).

The example solution we have seen demonstrates how to solve a simple first-order differential equation by using separation of variables. Such equations are the starting point for learning more complex differential equations, and they arise frequently in various scientific and mathematical applications—from physics to engineering to biology. Their solutions can be straightforward or quite complex, and may require special methods or considerations based on their structure.