Answer:
Let's complete the square first.
y = x² + 6x + 3
= (x² + 6x + 9) - 6
= (x + 3)² - 6
Therefore, the vertex is (-3, -6) and since the coefficient of (x + 3)² is positive, the vertex is a minimum.
Let , so that :
Now the ODE is separable, and we have
Integrating both sides gives
For the integral on the left, rewrite the integrand as
Then
and so
Given that , we find
so that the particular solution to this IVP is