Solving The Hyperbolic Equation 2cosh(2x) + 10sinh(2x) = 5
Introduction to Hyperbolic Functions
When delving into the world of hyperbolic equations, understanding the fundamental hyperbolic functions is crucial. Hyperbolic functions, denoted as sinh(x), cosh(x), tanh(x), coth(x), sech(x), and csch(x), are counterparts to the trigonometric functions but are defined using exponential functions. Specifically, the hyperbolic sine (sinh) and hyperbolic cosine (cosh) are defined as follows:
- sinh(x) = (e^x - e^-x) / 2
- cosh(x) = (e^x + e^-x) / 2
These functions exhibit unique properties and relationships that are essential when solving equations involving them. For instance, a key identity that connects sinh(x) and cosh(x) is:
cosh²(x) - sinh²(x) = 1
This identity is analogous to the trigonometric identity sin²(x) + cos²(x) = 1 and plays a significant role in simplifying and solving hyperbolic equations. The hyperbolic tangent, tanh(x), is another important function, defined as the ratio of sinh(x) to cosh(x):
tanh(x) = sinh(x) / cosh(x) = (e^x - e^-x) / (e^x + e^-x)
Understanding these definitions and identities is the first step in tackling hyperbolic equations. In the context of solving the equation 2cosh(2x) + 10sinh(2x) = 5, these foundational concepts will be instrumental in transforming the equation into a more manageable form. Grasping the interplay between exponential functions and hyperbolic functions allows for strategic manipulation and simplification, ultimately leading to the solution. Furthermore, recognizing the relationships between different hyperbolic functions, such as the identity cosh²(x) - sinh²(x) = 1, provides powerful tools for rewriting equations and isolating variables. As we proceed to solve the given equation, we will leverage these fundamental concepts to navigate the complexities and arrive at a clear and accurate solution.
Problem Statement: 2cosh(2x) + 10sinh(2x) = 5
The problem at hand involves solving the hyperbolic equation 2cosh(2x) + 10sinh(2x) = 5. This equation combines the hyperbolic cosine and hyperbolic sine functions, making it a non-trivial problem that requires careful manipulation and application of hyperbolic identities. To effectively solve this equation, we will employ a series of steps that leverage the definitions of cosh(2x) and sinh(2x) in terms of exponential functions. The primary goal is to transform the equation into a form that is easier to solve, typically a quadratic equation or an exponential equation that can be addressed using standard algebraic techniques.
The initial challenge lies in the presence of both cosh(2x) and sinh(2x) terms. To streamline the equation, we will substitute their exponential definitions: cosh(2x) = (e^(2x) + e^(-2x)) / 2 and sinh(2x) = (e^(2x) - e^(-2x)) / 2. This substitution allows us to rewrite the equation in terms of exponential functions, which can then be manipulated using algebraic methods. The resulting equation will likely involve exponential terms that need to be simplified and rearranged. For example, we might need to multiply through by a common denominator or combine like terms to create a more manageable expression. Furthermore, we will need to make strategic substitutions to further simplify the equation. A common approach is to introduce a new variable, such as y = e^(2x), which transforms the equation into a quadratic form. This substitution is particularly useful because it allows us to apply familiar methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula.
By transforming the original hyperbolic equation into a quadratic equation, we can leverage well-established techniques to find the values of the new variable. Once we have solved for the new variable, we will need to reverse the substitution to find the values of x. This involves taking the natural logarithm of both sides of the equation, which will isolate x. The final step is to check the solutions to ensure they are valid and satisfy the original equation. This is a crucial step because sometimes extraneous solutions can arise during the solving process. In summary, solving the equation 2cosh(2x) + 10sinh(2x) = 5 requires a combination of understanding hyperbolic function definitions, algebraic manipulation, strategic substitution, and careful verification of solutions. The following sections will detail each step of this process, providing a comprehensive solution to the problem.
Step-by-Step Solution
To solve the hyperbolic equation 2cosh(2x) + 10sinh(2x) = 5, we begin by substituting the exponential definitions of cosh(2x) and sinh(2x). Recall that:
- cosh(2x) = (e^(2x) + e^(-2x)) / 2
- sinh(2x) = (e^(2x) - e^(-2x)) / 2
Substituting these definitions into the equation gives us:
2 * [(e^(2x) + e^(-2x)) / 2] + 10 * [(e^(2x) - e^(-2x)) / 2] = 5
Simplifying the equation, we get:
(e^(2x) + e^(-2x)) + 5(e^(2x) - e^(-2x)) = 5
Now, distribute the 5:
e^(2x) + e^(-2x) + 5e^(2x) - 5e^(-2x) = 5
Combine like terms:
6e^(2x) - 4e^(-2x) = 5
To eliminate the negative exponent, multiply the entire equation by e^(2x):
6(e(2x))2 - 4 = 5e^(2x)
Rearrange the equation into a quadratic form:
6(e(2x))2 - 5e^(2x) - 4 = 0
Now, let y = e^(2x). The equation becomes:
6y^2 - 5y - 4 = 0
This is a quadratic equation in terms of y. We can solve this using the quadratic formula, factoring, or other methods. In this case, let's use the quadratic formula:
y = [-b ± sqrt(b^2 - 4ac)] / (2a)
Where a = 6, b = -5, and c = -4. Plugging in these values, we get:
y = [5 ± sqrt((-5)^2 - 4 * 6 * (-4))] / (2 * 6) y = [5 ± sqrt(25 + 96)] / 12 y = [5 ± sqrt(121)] / 12 y = [5 ± 11] / 12
This gives us two possible values for y:
y1 = (5 + 11) / 12 = 16 / 12 = 4 / 3 y2 = (5 - 11) / 12 = -6 / 12 = -1 / 2
Since y = e^(2x), and e^(2x) is always positive, we discard the negative solution y2 = -1/2. Thus, we have:
e^(2x) = 4 / 3
To solve for x, take the natural logarithm of both sides:
2x = ln(4 / 3) x = (1 / 2) * ln(4 / 3)
So, the solution to the hyperbolic equation is:
x = (1 / 2) * ln(4 / 3)
This solution represents the value of x that satisfies the original equation. By meticulously applying the definitions of hyperbolic functions, transforming the equation into a quadratic form, and solving for the variable, we have successfully found the solution. The next section will verify this solution to ensure its correctness.
Verification of the Solution
After obtaining the solution x = (1 / 2) * ln(4 / 3) for the hyperbolic equation 2cosh(2x) + 10sinh(2x) = 5, it is crucial to verify its correctness. Verification involves substituting the solution back into the original equation to confirm that it satisfies the equality. This step is essential to ensure that no errors were made during the solving process and that the solution is valid.
Substitute x = (1 / 2) * ln(4 / 3) into the original equation:
2cosh(2 * (1 / 2) * ln(4 / 3)) + 10sinh(2 * (1 / 2) * ln(4 / 3)) = 5
Simplify the expression:
2cosh(ln(4 / 3)) + 10sinh(ln(4 / 3)) = 5
Now, use the exponential definitions of cosh and sinh:
cosh(ln(4 / 3)) = (e^(ln(4 / 3)) + e^(-ln(4 / 3))) / 2
sinh(ln(4 / 3)) = (e^(ln(4 / 3)) - e^(-ln(4 / 3))) / 2
Recall that e^(ln(x)) = x and e^(-ln(x)) = 1 / x. Therefore:
e^(ln(4 / 3)) = 4 / 3 e^(-ln(4 / 3)) = 3 / 4
Substitute these values back into the expressions for cosh and sinh:
cosh(ln(4 / 3)) = ((4 / 3) + (3 / 4)) / 2 = (16 / 12 + 9 / 12) / 2 = (25 / 12) / 2 = 25 / 24
sinh(ln(4 / 3)) = ((4 / 3) - (3 / 4)) / 2 = (16 / 12 - 9 / 12) / 2 = (7 / 12) / 2 = 7 / 24
Now, substitute these values back into the equation:
2 * (25 / 24) + 10 * (7 / 24) = 5
Simplify the expression:
50 / 24 + 70 / 24 = 5
120 / 24 = 5
5 = 5
The equation holds true, which confirms that the solution x = (1 / 2) * ln(4 / 3) is correct. This verification step demonstrates the accuracy of our step-by-step solution process. By substituting the solution back into the original equation and verifying that the equality holds, we can be confident in the correctness of our result. In summary, solving hyperbolic equations requires a combination of understanding the definitions of hyperbolic functions, careful algebraic manipulation, and thorough verification of solutions. The next section will provide a summary of the entire process and highlight key steps and considerations.
Summary and Conclusion
In this article, we have successfully solved the hyperbolic equation 2cosh(2x) + 10sinh(2x) = 5. The process involved several key steps, starting with understanding the definitions of hyperbolic functions and their properties. We began by recognizing the exponential definitions of cosh(2x) and sinh(2x):
- cosh(2x) = (e^(2x) + e^(-2x)) / 2
- sinh(2x) = (e^(2x) - e^(-2x)) / 2
Substituting these definitions into the original equation allowed us to transform the problem into an algebraic equation involving exponential terms. This is a common and effective strategy for solving hyperbolic equations, as it leverages the fundamental relationship between hyperbolic functions and exponential functions.
Next, we simplified the equation by combining like terms and eliminating the negative exponent. This involved multiplying the entire equation by e^(2x), which resulted in a quadratic-like expression. To make the equation more manageable, we introduced a substitution, letting y = e^(2x). This transformed the equation into a quadratic equation in terms of y:
6y^2 - 5y - 4 = 0
We then solved this quadratic equation using the quadratic formula, which yielded two possible values for y. However, since y = e^(2x) must be positive, we discarded the negative solution and focused on the positive solution:
y = 4 / 3
Substituting back, we had:
e^(2x) = 4 / 3
Taking the natural logarithm of both sides allowed us to solve for x:
2x = ln(4 / 3) x = (1 / 2) * ln(4 / 3)
To ensure the correctness of our solution, we performed a crucial verification step. This involved substituting the solution x = (1 / 2) * ln(4 / 3) back into the original hyperbolic equation. By simplifying the expression and using the exponential definitions of cosh and sinh, we confirmed that the equation held true, thus validating our solution.
In conclusion, solving hyperbolic equations requires a systematic approach that combines understanding the definitions and properties of hyperbolic functions, algebraic manipulation, strategic substitutions, and thorough verification. The steps outlined in this article provide a comprehensive framework for tackling such problems. By mastering these techniques, one can confidently solve a wide range of hyperbolic equations and apply these skills in various mathematical and engineering contexts.
