Final answer:
After integrating dy/dx = 3x + 5, and applying the initial condition y = 6 when x = 0, we find that the final equation for y is y = (3/2)x^2 + 5x + 6.
Step-by-step explanation:
To solve the mathematical problem completely, we are given that dy/dx equals 3x + 5 and that the value of y is 6 when x is 0.
We can integrate the differential equation to find y.
- Integrate 3x + 5 with respect to x to get the general formula for y. The integral of 3x is (3/2)x^2, and the integral of 5 is 5x, so we get y = (3/2)x^2 + 5x + C, where C is the integration constant.
- Since we know that y = 6 when x = 0, we can substitute these values into our equation to find C. This gives us 6 = (3/2)·0^2 + 5·0 + C, which simplifies to C = 6.
- Therefore, the final equation for y is y = (3/2)x^2 + 5x + 6.