Answer is: option2
\( x^3 - x - 1 \)Solution:
We are given the differential equation:
\( \frac{dy}{dx} = 3x^2 - 1 \)
and the initial condition:
\( y = -1 \) when \( x = 1 \)
\( \frac{dy}{dx} = 3x^2 - 1 \)
Integrating both sides with respect to \( x \):
\[ y = \int (3x^2 - 1)\, dx = x^3 - x + C \]
Given \( y = -1 \) when \( x = 1 \):
\[ -1 = (1)^3 - 1 + C \Rightarrow -1 = 1 - 1 + C \Rightarrow -1 = C \]
\[ y = x^3 - x - 1 \]
Final Answer: (B) \( x^3 - x - 1 \)