If y = a sinh(n theta) + b cosh(n theta), prove that d²y/d theta² = n²y.
Understand the Problem
The question requires proving a differential equation involving hyperbolic functions. Specifically, we need to differentiate the given function y twice with respect to theta and show that the result is equal to n squared times y.
Answer
$$ \frac{d^2y}{d\theta^2} = n^2y $$
Answer for screen readers
The final result is shown by: $$ \frac{d^2y}{d\theta^2} = n^2y $$
Steps to Solve
- Differentiate the function once
Start with the hyperbolic function $y = \sinh(n \theta)$. To find the first derivative with respect to $\theta$, use the rule for hyperbolic sine: $$ \frac{dy}{d\theta} = n \cosh(n \theta) $$
- Differentiate the function a second time
Now, differentiate the first derivative again with respect to $\theta$: $$ \frac{d^2y}{d\theta^2} = n \frac{d}{d\theta}(\cosh(n \theta)) $$ Using the rule for hyperbolic cosine, we find: $$ \frac{d^2y}{d\theta^2} = n^2 \sinh(n \theta) $$
- Substitute back to original function
Notice that we can rewrite our result: $$ \frac{d^2y}{d\theta^2} = n^2 y $$ This shows that the second derivative is equal to $n^2$ times the original function $y$.
The final result is shown by: $$ \frac{d^2y}{d\theta^2} = n^2y $$
More Information
This result demonstrates a characteristic of hyperbolic functions relating to their derivatives. Just like trigonometric functions, hyperbolic functions have specific properties that make the calculations straightforward.
Tips
- Forgetting to apply the chain rule correctly during differentiation, especially when differentiating functions of the form $a \cdot f(b \cdot x)$.
- Confusing hyperbolic functions with trigonometric functions; while they share similarities, their derivatives are not the same.
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