Check if \( y = x^3 - 5x + 2 \) satisfies the differential equation \( \frac{dy}{dx} = 3x^2 - 5 \).

To check if the function \( y = x^3 - 5x + 2 \) satisfies the differential equation \( \frac{dy}{dx} = 3x^2 - 5 \), we first need to compute the derivative of \( y \) with respect to \( x \). Starting with the function: \[ y = x^3 - 5x + 2 \] we differentiate \( y \) with respect to \( x \): \[ \frac{dy}{dx} = \frac{d}{dx}(x^3) - \frac{d}{dx}(5x) + \frac{d}{dx}(2) \] Calculating each term: 1. The derivative of \( x^3 \) is \( 3x^2 \). 2. The derivative of \( -5x \) is \( -5 \). 3. The derivative of the constant \( 2 \) is \( 0 \). Putting it all together, we have: \[ \frac{dy}{dx} = 3x^2 - 5 \] Now, we compare this result with the right-hand side of the differential equation: \[ \frac{dy}{dx} = 3x^2 - 5 \] Since both sides are equal, we conclude that the function \( y = x^3 - 5x + 2 \) does indeed satisfy the differential equation \( \frac{dy}{dx} = 3x^2 - 5 \). Thus, the answer is: \[ \text{Yes, } y = x^3 - 5x + 2 \text{ satisfies the differential equation.} \]