Question

For the following exercises, use this scenario: A cable hanging under its own weight has a slope \(S=d y / d x\) that satisfies \(d S / d x=c \sqrt{1+S^{2}} .\) The constant \(c\) is the ratio of cable density to tension.Integrate \(d y / d x=\sinh (c x)\) to find the cable height \(y(x)\) if \(y(0)=1 / c\).

Short answer

The cable height is \( y(x) = \frac{1}{c} \cosh(cx) \).

Step-by-step solution

Integrate Slope Equation

We start with the equation \( \frac{d y}{d x} = \sinh(cx) \). We need to integrate this equation with respect to \( x \) to find \( y(x) \).

Apply Hyperbolic Identity

Recall that \( \sinh(x) = \frac{e^x - e^{-x}}{2} \). Thus, we need to integrate \( \sinh(cx) \) to get \( y(x) \).

Solve the Integration

The integral of \( \sinh(cx) \) with respect to \( x \) is \( \frac{1}{c} \cosh(cx) + C \), where \( C \) is the constant of integration.

Apply Initial Condition

We know \( y(0) = \frac{1}{c} \). At \( x = 0 \), substituting in the expression from the integration, we have: \( \frac{1}{c} \cosh(0) + C = \frac{1}{c} \). Since \( \cosh(0) = 1 \), we find \( C = 0 \).

Find the Final Expression for y(x)

Substitute \( C = 0 \) back into the expression from the integration. We get \( y(x) = \frac{1}{c} \cosh(cx) \) as the expression for the cable height \( y(x) \).

Key concepts

Hyperbolic Functions

Hyperbolic functions have similarities to the more well-known trigonometric functions, but they relate to hyperbolas rather than circles. One of the key hyperbolic functions is the hyperbolic sine, denoted as \( \sinh(x) \), which is defined by the formula \( \sinh(x) = \frac{e^x - e^{-x}}{2} \). This is quite useful in differential equations, especially when modeling physics scenarios like the catenary curve of a hanging cable.

In the given problem, the slope of a cable is expressed using \( \sinh(cx) \). This shows that hyperbolic functions play an integral role in understanding how such a curve behaves, especially under the influence of gravitational forces acting along with tension. Hyperbolic functions are instrumental because they can take into account various aspects like exponential growth and symmetrical properties inherent in such systems.

Integration Techniques

Integration is a fundamental technique in calculus used to find functions given their derivatives. In this exercise, integrating the slope equation \( \frac{d y}{d x} = \sinh(cx) \) allows us to find the function \( y(x) \).

We start by using the identity of the hyperbolic sine function to identify that we need to integrate \( \sinh(cx) \). The integration process involves finding the antiderivative, which results in \( \int \sinh(cx) \, dx = \frac{1}{c} \cosh(cx) + C \). Here, \( C \) is the constant of integration, which always appears due to the indefinite integral. Remember that constants like \( c \) should be treated carefully, as they often relate to real-world quantities such as the cable’s density and tension in our problem.

This technique not only helps solve problems involving hyperbolic functions but also extends to other contexts where integration by recognizing function identities is useful.

Initial Conditions

Initial conditions are crucial when solving differential equations because they provide the necessary information to determine the specific solution for a particular problem. In differential equations, solutions generally include an arbitrary constant because of the nature of indefinite integration.

In our problem, we have the initial condition \( y(0) = \frac{1}{c} \). Applying this condition allows us to determine the constant \( C \) gathered from the integration step. Substituting into \( \frac{1}{c} \cosh(0) + C = \frac{1}{c} \), we leverage the fact that \( \cosh(0) = 1 \). This simplifies to \( \frac{1}{c} + C = \frac{1}{c} \), revealing that \( C = 0 \).

By applying initial conditions, we ensure that the mathematical solution reflects the physical reality at a specific point, leading to the equation \( y(x) = \frac{1}{c} \cosh(cx) \) that accurately models the cable’s height based on the given scenario.