Final answer:
To solve the given initial value problem, we'll use the method of integrating factors. After rearranging the equation to standard form, we'll find the integrating factor and proceed to solve for y.
Step-by-step explanation:
To find the solution of the given initial value problem, we'll use the method of integrating factors. First, we'll rearrange the equation to the standard form: y' + (6/t)y = t - 1/t + 1. Now, let's identify the integrating factor, which is e^(∫6/t dt) = e^(6ln|t|) = t^6. Multiply both sides of the equation with t^6 to get t^7y' + 6t^5y = t^7 - t^6 + t^6. Now, we can integrate both sides and solve for y.