STATEMENT-1 : The differential equation `(d^(2)y)/(dx^(2)) + cos x.(dy)/(dx) + (x^(3) + 7)y = e^(x)` is a linear equation
and
STATEMENT-2 : Every first degree equation is a linear equation.

Options

A. Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is true, Statement-2 is true, Statement-2 is NOT a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To analyze the given statements, we need to evaluate each statement separately.

### Step 1: Analyze Statement 1
We have the differential equation:
\[ \frac{d^2y}{dx^2} + \cos(x) \frac{dy}{dx} + (x^3 + 7)y = e^x \]
To determine if this is a linear differential equation, we need to check the following criteria:
- The dependent variable \(y\) and its derivatives appear to the first power.
- There are no products or nonlinear functions of \(y\) or its derivatives.

In our equation:
- The term \(\frac{d^2y}{dx^2}\) is the second derivative of \(y\) and appears to the first power.
- The term \(\cos(x) \frac{dy}{dx}\) involves the first derivative of \(y\) and is multiplied by a function of \(x\), which is acceptable.
- The term \((x^3 + 7)y\) is a linear function of \(y\) since \(y\) is to the first power.
- The right-hand side \(e^x\) is a function of \(x\) and does not affect the linearity of the left side.

Since all terms involving \(y\) and its derivatives are linear, we conclude that Statement 1 is **true**.

### Step 2: Analyze Statement 2
Statement 2 claims that "Every first degree equation is a linear equation."

To clarify this, we need to understand the definitions:
- A first-degree equation in the context of differential equations refers to the highest derivative having a power of 1.
- A linear equation means that the dependent variable and its derivatives appear only to the first power and are not multiplied together or raised to any power other than 1.

To illustrate that Statement 2 is not necessarily true, consider the following example:
\[ \frac{d^2y}{dx^2} + \sin(x) \left(\frac{dy}{dx}\right)^2 + 3y = e^x \]
In this case:
- The highest derivative \(\frac{d^2y}{dx^2}\) is of degree 1.
- However, the term \(\left(\frac{dy}{dx}\right)^2\) is nonlinear because it involves the square of the first derivative.

Thus, even though the highest derivative is of degree 1, the presence of a nonlinear term means that this is not a linear equation. Therefore, Statement 2 is **false**.

### Conclusion
- **Statement 1** is true: The given differential equation is linear.
- **Statement 2** is false: Not every first-degree equation is a linear equation.