91Ó°ÊÓ

In differential equations, verifying that a function is truly a solution to a given equation is crucial. This process involves ensuring that when you substitute your proposed solution into the equation, both sides of the equation match perfectly.
In our exercise, we checked whether the function \( y = \sin t \) satisfies the differential equation \( \left(\frac{dy}{dt}\right)^{2} = 1-y^{2} \). By differentiating \( y \), calculating its derivative squared, and substituting back into the original equation, we compared both sides. They matched, thus confirming \( y = \sin t \) as a valid solution.
This procedure of substituting and verifying each side provides a logical and rigorous check to ensure a solution's validity.

Calculating derivatives is a fundamental skill in solving differential equations. It allows us to understand how a function changes with respect to a variable.
For the function \( y = \sin t \), the derivative with respect to \( t \) is \( \frac{dy}{dt} = \cos t \). This simple process involves applying known differentiation rules to find the rate of change of \( y \) with respect to \( t \).
Derivatives are pivotal as they transform functions into new expressions that help us analyze and predict the behavior of equations. In our problem, squaring the derivative, \( (\cos t)^2 \), allowed us to form a crucial component necessary for comparing both sides of the given equation.

Trigonometric identities are mathematical tools that relate the angles and ratios in trigonometry. They simplify complex expressions into more manageable forms. Probably the most famous one, \( \sin^2 t + \cos^2 t = 1 \), was critical in verifying our solution.
In our differential equation problem, substituting \( y = \sin t \) into \( 1 - y^2 \) was simplified using the identity to \( \cos^2 t \). This identity helped bridge the connection between the left and right sides of the equation, ensuring they are equivalent.
Understanding and applying these identities allows for the seamless solving of trigonometric-related differential equations. They often serve as shortcuts that can make complex calculations straightforward.