Consider the following differential equation: t^2\frac{d^{2}y}{d{t}^2}+7t\frac{dy}{dt}+8ty=10\sin \mathrm{(t)}..Which one of the following options is correct?

The correct option is It is a linear differential equation.

To determine why this is correct, let us analyze the properties of the given differential equation:

$$t^2\frac{d^{2}y}{d{t}^2}+7t\frac{dy}{dt}+8ty=10\sin \left(t\right)$$

A differential equation is defined as linear if it satisfies the following two conditions regarding the dependent variable (in this case, y) and its derivatives:
1. The dependent variable y and all its derivatives ($\frac{dy}{dt}$, $\frac{d^{2}y}{d{t}^2}$, etc.) occur only to the first power.
2. There are no products of the dependent variable y and its derivatives, nor are there nonlinear functions of y (such as sin(y), e^y, or y²).

Let us examine each term in the given equation:
- The term $t^2\frac{d^{2}y}{d{t}^2}$ features the second derivative $\frac{d^{2}y}{d{t}^2}$ to the first power. The coefficient is t², which depends only on the independent variable t.
- The term $7t\frac{dy}{dt}$ features the first derivative $\frac{dy}{dt}$ to the first power. The coefficient is 7t, which also depends only on t.
- The term 8ty features the dependent variable y to the first power with a coefficient of 8t.
- The right-hand side term, 10 sin(t), is a function only of the independent variable t.

Since the dependent variable y and its derivatives appear linearly (raised only to the power of 1, and not multiplied together), the differential equation is linear.

Why the other options are incorrect:
- "It is a nonlinear differential equation": Incorrect, because it satisfies all the conditions for linearity described above.
- "It is a time-invariant differential equation": Incorrect. A differential equation is time-invariant (or has constant coefficients) if the coefficients of y and its derivatives do not depend on the independent variable t. Here, the coefficients (t², 7t, 8t) are explicit functions of t, making it a time-varying (or variable-coefficient) equation.
- "It is a second-order partial differential equation": Incorrect. The derivatives in the equation are ordinary derivatives ($d$ instead of $\partial$) with respect to a single independent variable t. Therefore, it is an Ordinary Differential Equation (ODE), not a Partial Differential Equation (PDE).