Hyperbolic Functions, Exponential Change, and Separable Differential Equations

7.7 Hyperbolic Functions

Introduction to Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but are defined using exponential functions. They appear frequently in mathematics, physics, and engineering, especially in contexts involving hyperbolas, complex numbers, and certain differential equations.

• Key Hyperbolic Functions: sinh x (hyperbolic sine), cosh x (hyperbolic cosine), tanh x (hyperbolic tangent), coth x (hyperbolic cotangent), sech x (hyperbolic secant), csch x (hyperbolic cosecant).

• Applications: Used in modeling catenary curves, describing the shape of hanging cables, and in the design of structures such as the Gateway Arch in St. Louis.

• Similarity to Trigonometric Functions: Hyperbolic functions share many properties and identities with trigonometric functions, but are based on exponential rather than circular geometry.

Definitions and Formulas

The six basic hyperbolic functions are defined as follows:

• Hyperbolic Sine: $\sinh x = \frac{e^x - e^{-x}}{2}$

• Hyperbolic Cosine: $\cosh x = \frac{e^x + e^{-x}}{2}$

• Hyperbolic Tangent: $\tanh x = \frac{\sinh x}{\cosh x}$

• Hyperbolic Cotangent: $\coth x = \frac{1}{\tanh x}$, $x \neq 0$

• Hyperbolic Secant: $\sech x = \frac{1}{\cosh x}$

• Hyperbolic Cosecant: $\csch x = \frac{1}{\sinh x}$, $x \neq 0$

Graphs and Properties

Each hyperbolic function has a distinct graph and domain/range:

• $\sinh x$: Odd function, unbounded, passes through the origin.

• $\cosh x$: Even function, always positive, minimum at $x=0$.

• $\tanh x$: Odd function, horizontal asymptotes at $y=1$ and $y=-1$.

Example: The Gateway Arch in St. Louis is modeled using the hyperbolic cosine function $y = a \cosh(bx)$.

Identities of Hyperbolic Functions

Hyperbolic functions satisfy several important identities, similar to trigonometric identities:

• $\cosh^2 x - \sinh^2 x = 1$

• $\sinh(x + y) = \sinh x \cosh y + \cosh x \sinh y$

• $\cosh(x + y) = \cosh x \cosh y + \sinh x \sinh y$

• $\sinh 2x = 2 \sinh x \cosh x$

• $\cosh 2x = \cosh^2 x + \sinh^2 x$

• $\tanh^2 x = 1 - \sech^2 x$

• $\coth^2 x = 1 + \csch^2 x$

Derivatives and Integrals of Hyperbolic Functions

The derivatives and integrals of hyperbolic functions are straightforward and closely resemble those of trigonometric functions.

• $\frac{d}{dx} \sinh x = \cosh x$

• $\frac{d}{dx} \cosh x = \sinh x$

• $\frac{d}{dx} \tanh x = \sech^2 x$

• $\frac{d}{dx} \coth x = -\csch^2 x$

• $\frac{d}{dx} \sech x = -\sech x \tanh x$

• $\frac{d}{dx} \csch x = -\csch x \coth x$

Integral Formulas:

• $\int \sinh x \, dx = \cosh x + C$

• $\int \cosh x \, dx = \sinh x + C$

• $\int \tanh x \, dx = \ln|\cosh x| + C$

• $\int \coth x \, dx = \ln|\sinh x| + C$

• $\int \sech x \, dx = 2 \arctan(\tanh \frac{x}{2}) + C$

• $\int \csch x \, dx = \ln|\tanh \frac{x}{2}| + C$

Inverse Hyperbolic Functions

Definitions and Properties

Inverse hyperbolic functions are the inverses of the basic hyperbolic functions. They are useful for solving equations involving hyperbolic functions and for evaluating certain integrals.

• Inverse Hyperbolic Sine: $\sinh^{-1} x = \ln(x + \sqrt{x^2 + 1})$

• Inverse Hyperbolic Cosine: $\cosh^{-1} x = \ln(x + \sqrt{x^2 - 1}), \quad x \geq 1$

• Inverse Hyperbolic Tangent: $\tanh^{-1} x = \frac{1}{2} \ln\left(\frac{1 + x}{1 - x}\right), \quad |x| < 1$

• Inverse Hyperbolic Cotangent: $\coth^{-1} x = \frac{1}{2} \ln\left(\frac{x + 1}{x - 1}\right), \quad |x| > 1$

• Inverse Hyperbolic Secant: $\sech^{-1} x = \ln\left(\frac{1}{x} + \sqrt{\frac{1}{x^2} - 1}\right), \quad 0 < x \leq 1$

• Inverse Hyperbolic Cosecant: $\csch^{-1} x = \ln\left(\frac{1}{x} + \sqrt{\frac{1}{x^2} + 1}\right), \quad x \neq 0$

Derivatives of Inverse Hyperbolic Functions

• $\frac{d}{dx} \sinh^{-1} x = \frac{1}{\sqrt{x^2 + 1}}$

• $\frac{d}{dx} \cosh^{-1} x = \frac{1}{\sqrt{x^2 - 1}}, \quad x > 1$

• $\frac{d}{dx} \tanh^{-1} x = \frac{1}{1 - x^2}, \quad |x| < 1$

• $\frac{d}{dx} \coth^{-1} x = \frac{1}{1 - x^2}, \quad |x| > 1$

• $\frac{d}{dx} \sech^{-1} x = -\frac{1}{x \sqrt{1 - x^2}}, \quad 0 < x < 1$

• $\frac{d}{dx} \csch^{-1} x = -\frac{1}{|x| \sqrt{1 + x^2}}, \quad x \neq 0$

Integrals Leading to Inverse Hyperbolic Functions

Some integrals can be evaluated using inverse hyperbolic functions:

• $\int \frac{1}{\sqrt{x^2 + a^2}} \, dx = \sinh^{-1}\left(\frac{x}{a}\right) + C$

• $\int \frac{1}{\sqrt{x^2 - a^2}} \, dx = \cosh^{-1}\left(\frac{x}{a}\right) + C$

• $\int \frac{1}{a^2 - x^2} \, dx = \frac{1}{a} \tanh^{-1}\left(\frac{x}{a}\right) + C$

Exponential Change and Separable Differential Equations

Exponential Change

Exponential change describes processes where the rate of change of a quantity is proportional to its current value. This is common in population growth, radioactive decay, and cooling/heating problems.

• General Form: $\frac{dy}{dt} = k y$

• Solution: $y = y_0 e^{kt}$, where $y_0$ is the initial value and $k$ is the rate constant.

• Exponential Growth: $k > 0$

• Exponential Decay: $k < 0$

Example: Atmospheric pressure decreases exponentially with altitude. Radioactive decay follows $y = y_0 e^{-kt}$, where $k$ is related to the half-life by $t_{1/2} = \frac{\ln 2}{k}$.

Newton's Law of Cooling

Newton's Law of Cooling states that the rate of change of temperature of an object is proportional to the difference between its temperature and the ambient temperature.

• Differential Equation: $\frac{dH}{dt} = -k(H - H_s)$, where $H$ is the object's temperature and $H_s$ is the surrounding temperature.

• Solution: $H(t) = H_s + (H_0 - H_s) e^{-kt}$

• Application: Used to model cooling of objects, such as an egg cooling in water.

Separable Differential Equations

A separable differential equation is one that can be written as a product of a function of $y$ and a function of $x$:

• General Form: $\frac{dy}{dx} = f(x)g(y)$

• Solution Method:

  1. Rewrite as $\frac{1}{g(y)} dy = f(x) dx$

  2. Integrate both sides: $\int \frac{1}{g(y)} dy = \int f(x) dx$

  3. Solve for $y$ if possible

• Example: $\frac{dy}{dx} = y(x+1)$ can be separated and solved by integrating both sides.

HTML Table: Hyperbolic Function Identities

Identity	Formula
Pythagorean Identity	$\cosh^2 x - \sinh^2 x = 1$
Double Angle for sinh	$\sinh 2x = 2 \sinh x \cosh x$
Double Angle for cosh	$\cosh 2x = \cosh^2 x + \sinh^2 x$
Sum Formula for sinh	$\sinh(x + y) = \sinh x \cosh y + \cosh x \sinh y$
Sum Formula for cosh	$\cosh(x + y) = \cosh x \cosh y + \sinh x \sinh y$

HTML Table: Derivatives of Inverse Hyperbolic Functions

Function	Derivative	Domain
$\sinh^{-1} x$	$\frac{1}{\sqrt{x^2 + 1}}$	All real $x$
$\cosh^{-1} x$	$\frac{1}{\sqrt{x^2 - 1}}$	$x > 1$
$\tanh^{-1} x$	$\frac{1}{1 - x^2}$	$|x| < 1$
$\coth^{-1} x$	$\frac{1}{1 - x^2}$	$|x| > 1$
$\sech^{-1} x$	$-\frac{1}{x \sqrt{1 - x^2}}$	$0 < x < 1$
$\csch^{-1} x$	$-\frac{1}{|x| \sqrt{1 + x^2}}$	$x \neq 0$

Summary

• Hyperbolic functions are defined using exponentials and have properties similar to trigonometric functions.

• Inverse hyperbolic functions are useful for solving equations and evaluating integrals.

• Exponential change models growth and decay processes, including population, radioactive decay, and cooling.

• Separable differential equations can be solved by separating variables and integrating both sides.