Hyperbolic Equations Calculator

Ask and get solution to your homeworkAsk now and get step-by-step solutions

Example

Created on 2024-06-20Asked by Noah Young (Solvelet student)

Solve the hyperbolic equation

\sinh(x) = 1

Solution

To solve the hyperbolic equation

\sinh(x) = 1

: The hyperbolic sine function is defined as:

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

Set this equal to 1:

\frac{e^x - e^{-x}}{2} = 1

Multiply both sides by 2:

e^x - e^{-x} = 2

Let

y = e^x

. Then

e^{-x} = \frac{1}{y}

, so:

y - \frac{1}{y} = 2

Multiply both sides by

y

y^2 - 1 = 2y

y^2 - 2y - 1 = 0

Solve this quadratic equation using the quadratic formula

y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

y = \frac{2 \pm \sqrt{4 + 4}}{2}

y = \frac{2 \pm \sqrt{8}}{2}

y = 1 \pm \sqrt{2}

Since

y = e^x

and

y

must be positive:

y = 1 + \sqrt{2}

Thus:

e^x = 1 + \sqrt{2}

Taking the natural logarithm of both sides:

x = \ln(1 + \sqrt{2})

Therefore, the solution to the hyperbolic equation

\sinh(x) = 1

is:

x = \ln(1 + \sqrt{2})

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Mia Williams on Solvelet

1. Solve the one-dimensional wave equation

\frac{\partial^2 u}{\partial t^2} - c^2 \frac{\partial^2 u}{\partial x^2} = 0

subject to the initial conditions

u(x, 0) = f(x)

and

\frac{\partial u}{\partial t}(x, 0) = g(x)

,2. Conduct a hypothesis test to determine whether the proportion of defective items in a manufacturing process is less than

0.10

, given a sample size of

200

and

20

defective items.,

DefinitionHyperbolic equations are a class of partial differential equations (PDEs) that have the properties of describing wave propagation and signal transmission. A key feature is that they come in a certain form that includes second-order derivatives, and their solutions look distinctly wavelike. Hyperbolic equations are used in physics, engineering, and other fields to model phenomena like sound waves, electromagnetic waves, and fluid dynamics. For example, the wave equation ∂t2∂2u=c2∇2u is a hyperbolic model for the wave-like propagation of waves in as to what medium, where u is the wave function and c is the speed of the waves.

Related Calculators

Bessel Functions Calculator Riemann Sums Calculator Range and Standard Deviation Calculator Ring Theory Calculator

Need topic explanation ? Get video explanation