Problem 56 - Calculus: Graphical, Numerical, Algebraic

Question

Perpendicular Slope Fields If the slope fields for the differ- ential equations \(d y / d x=\sec x\) and \(d y / d x=g(x)\) are perpendicu- lar (as in Exercise \(55 ),\) find \(g(x)\) .

Step-by-Step Solution

Answer

The function \(g(x)\) that describes the slope of the line perpendicular to the given line is \(g(x) = -\cos x\)

Step 1: Express the given Slope

We are provided with one slope \(dy/dx=\sec x\). We also know that for two lines to be perpendicular, the product of their slopes is -1.

Step 2: Determine the perpendicular Slope

If \(\sec x\) is the slope of one line, then the slope of the line which is perpendicular to it is therefore \(-1/\sec x\), because the product of the slopes of two perpendicular lines is -1.

Step 3: Simplify the expression

Note that \(\sec x\) is equivalent to \(1/\cos x \). Consequently, \(-1/\sec x\) becomes \(-1/(1/\cos x)\) which simplifies to \(-\cos x\). Hence, the function \(g(x)\) that describes the slope of this line is \(-\cos x\)

Key Concepts

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. In simple terms, they express relationships between changing quantities.
These equations are fundamental in describing various physical phenomena like motion, heat, electricity, and more.

• Basic Form: A common form is \(\frac{dy}{dx} = f(x)\), where \(y\) is a function of \(x\).
• Types: Differential equations can be ordinary (ODEs) when involving functions of a single variable, or partial (PDEs) when involving multiple variables.

When solving a differential equation, the goal is often to find a function \(y\) that satisfies the equation based on given conditions. For our exercise, we have a differential equation \(\frac{dy}{dx}=\sec x\).
Understanding how this relates to a second differential equation involves analyzing their slopes, as we will explore further in slope fields and perpendicular lines.

Slope Fields

Slope fields, also known as direction fields, provide a visual way to represent differential equations. They consist of small line segments or arrows that show how the slope of the solution curves varies from point to point.
This helps in understanding the behavior of solutions without actually solving the differential equation analytically.

• Construction: For a differential equation like \(\frac{dy}{dx} = \sec x\), plot the slope \(\sec x\) at various points in the plane.
• Interpretation: The line segments indicate the direction the solution would take at any point.

In our exercise, these slope fields help us understand how one differential equation compares to another having perpendicular slopes. This visual tool provides insight into the behavior of functions defined by differential equations, especially when slopes are perpendicular, as realized in perpendicular lines.

Perpendicular Lines

Perpendicular lines are lines that intersect to form right angles, precisely 90 degrees. In mathematics, when considering the slopes of two lines, their relationship is paramount in determining perpendicularity.
The rule is straightforward: if the product of the slopes of two lines is \(-1\), then they are perpendicular.

• Rule of Perpendicular Slopes: Suppose one line has a slope \(m_1\), then the slope \(m_2\) of a line perpendicular to it satisfies \(m_1 \times m_2 = -1\).
• Example: For our problem, given \(dy/dx = \sec x\), the perpendicular slope would be \(-\cos x\), as seen from \(-1/\sec x = -\cos x\).

Understanding the concept of perpendicularity in slope helps us derive the function \(g(x)\). For our exercise, \(g(x) = -\cos x\) such that the slopes will be perpendicular. This illustrates a key connection between geometry and calculus, providing a powerful way to solve differential equation problems.