Question

Find a second-order differential equation that is satisfied by $$ y=A \cosh 2 x+B \sinh 2 x $$

Short answer

Answer: The second-order differential equation satisfied by the given function is \(\frac{d^2y}{dx^2} = 4y\).

Step-by-step solution

Find the first derivative of the function

To find the first derivative of the function with respect to x, we apply the power and chain rule: $$ \frac{dy}{dx} = \frac{d}{dx}(A \cosh 2x + B \sinh 2x) = 2A \sinh 2x + 2B \cosh 2x $$

Find the second derivative of the function

Now, we find the second derivative of the function with respect to x by differentiating the first derivative: $$ \frac{d^2y}{dx^2} = \frac{d}{dx} (2A \sinh 2x + 2B \cosh 2x) = 4A \cosh 2x + 4B \sinh 2x $$

Find a relationship between the function and its derivatives

We can rewrite the second derivative by factoring out a 4 from both terms: $$ \frac{d^2y}{dx^2} = 4(A \cosh 2x + B \sinh 2x) $$ Notice that the expression inside the parentheses is the same as the original function, \(y\). Therefore, we have found a relationship between the original function and its second derivative: $$ \frac{d^2y}{dx^2} = 4y $$

Write the final answer

The second-order differential equation satisfied by the given function is $$ \frac{d^2y}{dx^2} = 4y $$

Key concepts

Differential Derivatives

Differential derivatives are essential tools in calculus that help us understand how functions change. A derivative measures the rate at which a function's value changes as its input changes. When dealing with second-order derivatives, like in the original exercise, we are interested in how the rate of change itself changes. This is crucial when solving differential equations. In our specific problem, we start with a given function.

• First Derivative: This involves applying basic rules, like the power rule or the chain rule, to find the rate of change of the function. For example, the first derivative of our function is determined to be \( 2A \sinh 2x + 2B \cosh 2x \).
• Second Derivative: This tells us how the first derivative, or the rate of change, is changing. Finding the second derivative requires taking the derivative again. Here, it results in \( 4A \cosh 2x + 4B \sinh 2x \).

Understanding differential derivatives allows us to form equations that predict the behavior of functions, crucial in fields like physics, engineering, and economics.

Hyperbolic Functions

Hyperbolic functions, such as hyperbolic sine (\(\sinh\)) and hyperbolic cosine (\(\cosh\)), are analogs of traditional trigonometric functions, but with differences that make them unique in calculus and differential equations. These functions appear in contexts where exponential functions are involved and are particularly useful in hyperbolic geometry and complex analysis.

• Definition: The hyperbolic sine and cosine are defined as \(\sinh x = \frac{e^x - e^{-x}}{2}\) and \(\cosh x = \frac{e^x + e^{-x}}{2}\).
• Properties: They satisfy identities similar to trigonometric functions, such as \(\cosh^2 x - \sinh^2 x = 1\).
• Derivatives: The derivatives of these functions are also straightforward: the derivative of \(\sinh x\) is \(\cosh x\), and the derivative of \(\cosh x\) is \(\sinh x\).

In our problem, these functions allow us to express the given function and its derivatives, leading to the second-order differential equation.

Chain Rule

The chain rule is an essential differentiation technique used in calculus to differentiate compositions of functions. It's a way of understanding how changes in one variable lead to changes in another when functions are nested.

• Basic Idea: If you have a composite function \(f(g(x))\), the derivative can be found by multiplying the derivative of \(f\) with respect to \(g(x)\) by the derivative of \(g(x)\) with respect to \(x\).
• Formula: Mathematically, it's expressed as \(\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)\).
• Application: In our exercise, the chain rule helps differentiate \(\cosh 2x\) and \(\sinh 2x\), as we have to account for the inner function \(2x\).

By using the chain rule, we correctly compute the derivatives needed to solve and verify the second-order differential equation. This rule is a foundational concept that appears in various calculus problems involving nested functions.