Question

Determine the conditions under which the equation $$ a \cosh x+b \sinh x=c, \quad c>0 $$ has zero, one, or two real solutions for \(x\). What is the solution if \(a^{2}=c^{2}+b^{2}\) ?

Short answer

For the equation to have zero, one, or two solutions depends on the discriminant of the quadratic equation derived from the original one. If \(a^2 = c^2 + b^2\), then there is exactly one real solution.

Step-by-step solution

Understand Hyperbolic Functions

Recall that the hyperbolic functions are defined as follows: \( \text{cosh}(x) = \frac{e^x + e^{-x}}{2} \) and \( \text{sinh}(x) = \frac{e^x - e^{-x}}{2} \)

Analysis of the given equation

Consider the given equation: \( a \text{cosh}(x) + b \text{sinh}(x) = c \). This can be rewritten using the definitions of hyperbolic functions.

Using the identity

Use the identity \( \text{cosh}^2(x) - \text{sinh}^2(x) = 1 \) to help simplify the expression.

Discriminant Analysis

Analyze the quadratic form of the equation to determine the conditions of zero, one, or two real solutions. Start by rewriting your equation influenced by hyperbolic identities.

Condition for zero, one, or two solutions

Determine the discriminant of the quadratic equation derived. A positive discriminant indicates two real solutions, zero discriminant indicates one real solution, and a negative discriminant indicates no real solution.

Special case analysis

Given that \(a^2 = c^2 + b^2\), substitute and solve for any specific or special conditions. Simplify the equation to identify solutions.

Solve the simplified equation

Isolate and solve for \(x\) to determine the explicit solutions or conditions.

Key concepts

hyperbolic identities

Hyperbolic functions, such as \text{cosh}(x) and \text{sinh}(x), are essential in solving many mathematical problems. These functions are defined as: \( \text{cosh}(x) = \frac{e^x + e^{-x}}{2} \) and \( \text{sinh}(x) = \frac{e^x - e^{-x}}{2} \). They resemble trigonometric functions but are based on exponential functions.
For instance, consider the equation in the problem: \( a \text{cosh}(x) + b \text{sinh}(x) = c \). It's crucial to understand that hyperbolic identities, like \( \text{cosh}^2(x) - \text{sinh}^2(x) = 1 \), can simplify this equation.
Let's use these identities to isolate variables. For example, we can rewrite the equation using the definitions provided, which will help us simplify and better analyze the given problem.

discriminant analysis

Discriminant analysis is a powerful tool to determine the number of real solutions in quadratic equations. For our problem, we need to convert the equation into a quadratic form.
Rewrite the equation using hyperbolic identities: \( a \text{cosh}(x) + b \text{sinh}(x) = c \). By simplifying, we can make it resemble a quadratic equation in the form of \( Ax^2 + Bx + C = 0 \).
To find out if we have zero, one, or two real solutions, we examine the discriminant: \( \text{Discriminant} = B^2 - 4AC \).

• If the discriminant is positive, there are two real solutions.
• If the discriminant equals zero, there is exactly one real solution.
• If the discriminant is negative, there are no real solutions.

With these criteria, we can analyze any quadratic equation derived from our original problem.

real solutions

To determine real solutions for the equation \( a \text{cosh}(x) + b \text{sinh}(x) = c \), we need to look at specific conditions derived from the equation. Consider the special case where \( a^2 = c^2 + b^2 \).
Substitute this condition into our equation. We get \( a \text{cosh}(x) + b \text{sinh}(x) = c \) and try to simplify it using known hyperbolic identities. By substituting \( a = \text{cosh}(y) \) and \( b = \text{sinh}(y) \), where \( y \) is some constant, we can often reduce the complexity of the problem.
Isolate \( x \) and solve the resulting simplified equation. This approach will help find the explicit real solutions or determine the conditions under which they exist.