Answer :
The sain Aolution to the initial value problem is = 3 cos(2) - 1 + 6sin(2) - 3sin(2) + 1.
In the initial value problem, we're given that the derivative of y with respect to x is equal to 1 plus 6 times x, and the value of y when x equals 3 is -3. To solve this, we'll integrate the derivative to find the function y(x), and then use the initial condition to determine the constant of integration.
Integrating the derivative 1 + 6x with respect to x gives us y(x) = x + 3x^2 + C, where C is the constant of integration. Now, using the initial condition y(3) = -3, we plug in x = 3 and y = -3 into our function to solve for C. This gives us -3 = 3 + 27 + C, so C = -33. Therefore, the solution to the initial value problem is y(x) = x + 3x^2 - 33.