The uploaded image contains a second-order differential equation along with a set of parametric functions. Here’s the breakdown of the equation and functions:

Differential Equation:

$\frac{d^2y}{dx^2}(x, y)^2 - 2 \left( x \frac{dy}{dx} \right)(x, y) \left( \frac{dy}{dx} \right) = 0$

Parametric Functions:

1. $x(t) = e^t \sin t$
2. $y(t) = e^t \cos t$

Interpretation:

The differential equation and the parametric functions suggest a relationship between $x$ and $y$ in terms of $t$. This can be interpreted as solving the differential equation by expressing $x$ and $y$ as functions of $t$.

Step 1: Derivatives from Parametric Equations

To analyze the differential equation, we'll need derivatives of $x(t)$ and $y(t)$:

1. First derivative: $\frac{dx}{dt} = \frac{d}{dt}(e^t \sin t) = e^t \sin t + e^t \cos t$ $\frac{dy}{dt} = \frac{d}{dt}(e^t \cos t) = e^t \cos t - e^t \sin t$

2. Second derivatives: $\frac{d^2x}{dt^2} = \frac{d}{dt}(e^t \sin t + e^t \cos t) = e^t \cos t + e^t \cos t - e^t \sin t = 2e^t \cos t$ $\frac{d^2y}{dt^2} = \frac{d}{dt}(e^t \cos t - e^t \sin t) = e^t \cos t - e^t \cos t - e^t \sin t - e^t \sin t = -2e^t \sin t$

Step 2: Expressing $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$

Using the chain rule: $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{e^t \cos t - e^t \sin t}{e^t \sin t + e^t \cos t}$ Simplify: $\frac{dy}{dx} = \frac{\cos t - \sin t}{\sin t + \cos t}$

For the second derivative: $\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$

The numerator involves differentiating $\frac{dy}{dx}$ with respect to $t$, which can be computed as a quotient rule.

Would you like me to compute the second derivative explicitly or verify if the parametric equations satisfy the given differential equation? Let me know! 😊

Follow-Up Questions:

1. How would you verify if the parametric solution satisfies the differential equation?
2. What is the geometric interpretation of the parametric curves defined by $x(t)$ and $y(t)$?
3. How do we compute higher-order derivatives for parametric equations efficiently?
4. Could you explain why solving the equation in terms of $t$ might simplify the analysis?
5. What methods can be used to solve second-order differential equations like this one?

Tip: For parametric equations, always confirm that derivatives are consistent with the chain rule before substitution into a differential equation!