If y = a cosh(x/a), prove that d²y/dx² = (1/a) * sqrt(1 + (dy/dx) ²)
Understand the Problem
The question requires us to prove a given differential equation. We are given (y = a \cosh(x/a)) and we need to show that (\frac{d^2y}{dx^2} = \frac{1}{a} \sqrt{1 + (\frac{dy}{dx})^2}). This involves finding the first and second derivatives of (y) with respect to (x), and then manipulating the expressions to arrive at the desired result.
Answer
$\frac{d^2y}{dx^2} = \frac{1}{a} \sqrt{1 + (\frac{dy}{dx})^2}$
Answer for screen readers
$\frac{d^2y}{dx^2} = \frac{1}{a} \sqrt{1 + (\frac{dy}{dx})^2}$
Steps to Solve
- Find the first derivative $\frac{dy}{dx}$
Given $y = a \cosh(\frac{x}{a})$, we differentiate with respect to $x$:
$\frac{dy}{dx} = a \cdot \sinh(\frac{x}{a}) \cdot \frac{1}{a} = \sinh(\frac{x}{a})$
- Find the second derivative $\frac{d^2y}{dx^2}$
Differentiate $\frac{dy}{dx} = \sinh(\frac{x}{a})$ with respect to $x$:
$\frac{d^2y}{dx^2} = \cosh(\frac{x}{a}) \cdot \frac{1}{a} = \frac{1}{a} \cosh(\frac{x}{a})$
- Express $\cosh(\frac{x}{a})$ in terms of $\sinh(\frac{x}{a})$
We know the identity $\cosh^2(u) - \sinh^2(u) = 1$. Therefore, $\cosh(u) = \sqrt{1 + \sinh^2(u)}$.
So, $\cosh(\frac{x}{a}) = \sqrt{1 + \sinh^2(\frac{x}{a})}$
- Substitute $\cosh(\frac{x}{a})$ into the second derivative expression
Substitute $\cosh(\frac{x}{a}) = \sqrt{1 + \sinh^2(\frac{x}{a})}$ into $\frac{d^2y}{dx^2} = \frac{1}{a} \cosh(\frac{x}{a})$:
$\frac{d^2y}{dx^2} = \frac{1}{a} \sqrt{1 + \sinh^2(\frac{x}{a})}$
- Substitute $\frac{dy}{dx}$ into the equation
Since $\frac{dy}{dx} = \sinh(\frac{x}{a})$, substitute this into the above equation:
$\frac{d^2y}{dx^2} = \frac{1}{a} \sqrt{1 + (\frac{dy}{dx})^2}$
Thus, we have shown that $\frac{d^2y}{dx^2} = \frac{1}{a} \sqrt{1 + (\frac{dy}{dx})^2}$.
$\frac{d^2y}{dx^2} = \frac{1}{a} \sqrt{1 + (\frac{dy}{dx})^2}$
More Information
The hyperbolic cosine function, denoted as $\cosh(x)$, is defined as $\cosh(x) = \frac{e^x + e^{-x}}{2}$. Its derivative is $\sinh(x)$, the hyperbolic sine function. The relationship $\cosh^2(x) - \sinh^2(x) = 1$ is a fundamental identity in hyperbolic trigonometry.
Tips
A common mistake is an error in the differentiation of the hyperbolic functions. It's important to remember that the derivative of $\cosh(x)$ is $\sinh(x)$ and the derivative of $\sinh(x)$ is $\cosh(x)$. Another mistake is incorrectly applying the chain rule. Also, errors can occur when manipulating the hyperbolic identities.
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