Problem 379 - Calculus Volume 1

Question

Show that \(\cosh (x)\) and \(\sinh (x)\) satisfy \(y^{\prime \prime}=y\)

Step-by-Step Solution

Answer

Both \(\cosh(x)\) and \(\sinh(x)\) satisfy the equation \(y'' = y\).

Step 1: Define Hyperbolic Functions

The hyperbolic cosine and sine functions are defined as follows:\[\cosh(x) = \frac{e^x + e^{-x}}{2}, \quad \sinh(x) = \frac{e^x - e^{-x}}{2}\]

Step 2: Differentiate \(\cosh(x)\)

Find the first derivative of \(\cosh(x)\) with respect to \(x\):\[\frac{d}{dx}\left(\cosh(x)\right) = \frac{d}{dx}\left(\frac{e^x + e^{-x}}{2}\right) = \frac{1}{2}(e^x - e^{-x}) = \sinh(x)\]

Step 3: Differentiate \(\sinh(x)\)

Find the first derivative of \(\sinh(x)\) with respect to \(x\):\[\frac{d}{dx}\left(\sinh(x)\right) = \frac{d}{dx}\left(\frac{e^x - e^{-x}}{2}\right) = \frac{1}{2}(e^x + e^{-x}) = \cosh(x)\]

Step 4: Verify Second Derivative of \(\cosh(x)\)

Differentiate \(\sinh(x)\), the first derivative of \(\cosh(x)\), to find its second derivative:\[\frac{d}{dx}\left(\sinh(x)\right) = \cosh(x)\]So, the second derivative of \(\cosh(x)\) is \(\cosh(x)\), which satisfies \(y'' = y\) for \(y = \cosh(x)\).

Step 5: Verify Second Derivative of \(\sinh(x)\)

Differentiate \(\cosh(x)\), the first derivative of \(\sinh(x)\), to find its second derivative:\[\frac{d}{dx}\left(\cosh(x)\right) = \sinh(x)\]So, the second derivative of \(\sinh(x)\) is \(\sinh(x)\), which satisfies \(y'' = y\) for \(y = \sinh(x)\).

Key Concepts

Understanding cosh(x)

The hyperbolic cosine function, denoted as \(\cosh(x)\), is a fundamental concept in hyperbolic functions. It resembles the ordinary cosine function but is instead designed for a hyperbola. The mathematical definition of \(\cosh(x)\) is:

• \(\cosh(x) = \frac{e^x + e^{-x}}{2}\)

This formula reveals that \(\cosh(x)\) is the average of the exponential function \(e^x\) and its reciprocal \(e^{-x}\). This symmetric nature around the y-axis makes it an even function.
When you take its derivative with respect to \(x\), you get \(\sinh(x)\), as its derivative reverses the sign of the exponential difference:

• \(\frac{d}{dx}(\cosh(x)) = \sinh(x)\)

Further differentiation brings \(\cosh(x)\) back as the result, proving that the second derivative of \(\cosh(x)\) returns \(\cosh(x)\) itself:

• \(\frac{d^2}{dx^2}(\cosh(x)) = \cosh(x)\)

This satisfies the differential equation \(y'' = y\), illustrating a key behavior of hyperbolic functions.

sinh(x) and Its Properties

The hyperbolic sine function, or \(\sinh(x)\), is another crucial hyperbolic function, part of the pair that includes \(\cosh(x)\). It's defined as follows:

• \(\sinh(x) = \frac{e^x - e^{-x}}{2}\)

The formula for \(\sinh(x)\) encapsulates the difference rather than the sum, symbolizing its odd nature similar to the sine function. The function \(\sinh(x)\) displays a natural symmetry about the origin, reflecting its identity as an odd function.
The first derivative of \(\sinh(x)\) with respect to \(x\) is \(\cosh(x)\), notably illustrating their interdependent relationship:

• \(\frac{d}{dx}(\sinh(x)) = \cosh(x)\)

Upon further differentiation, the second derivative returns \(\sinh(x)\), further confirming that \(\sinh(x)\) satisfies the same differential equation \(y'' = y\):

• \(\frac{d^2}{dx^2}(\sinh(x)) = \sinh(x)\)

This condition, \(y'' = y\), proves that \(\sinh(x)\) plays a vital role analogous to exponential growth and decay in real-world contexts.

Exploring the Second Derivative

Understanding the second derivative in the context of hyperbolic functions like \(\cosh(x)\) and \(\sinh(x)\) is essential to grasp their mathematical behavior. The second derivative represents the rate at which the first derivative itself changes, offering deeper insights into a function's curvature and graph behavior.
For hyperbolic functions:

• The second derivative of \(\cosh(x)\) is itself: \(\cosh(x)\).
• The second derivative of \(\sinh(x)\) is again itself: \(\sinh(x)\).

These results align perfectly with the differential equation \(y'' = y\), showcasing a cyclical pattern with these functions, where differentiation cyclically returns the original function. This pattern reveals their intrinsic stability and recurrence properties.
In practical applications, this suggests that changes in hyperbolic functions are consistent and predictable, much like the repeating exponential behavior found in natural phenomena. These functions offer robust models in engineering and physics, reflecting structures like catenary curves and describing the trajectory of objects under gravity. Recognizing these consistent mathematical properties can provide deeper understanding in various fields beyond pure mathematics.