Final answer:
To solve the given initial value problem (IVP), we can integrate both sides of the equation with respect to t. This will give us an expression for x(t). However, the integral of the right side of the equation is not elementary, so we cannot find an explicit solution for x(t).
Step-by-step explanation:
To solve the given initial value problem (IVP), we can integrate both sides of the equation with respect to t. This will give us an expression for x(t). Let's start by integrating the right side of the equation:
∫ (2x + 6) dx / dt = ∫ -t sin(t²) dt
The integral of (2x + 6) with respect to x is x² + 6x. The integral of -t sin(t²) with respect to t is not elementary, meaning it cannot be expressed in terms of elementary functions. Therefore, we cannot find an explicit solution for x(t). However, we can use numerical methods or approximation techniques to find an approximate solution.