Final answer:
To confirm that {1, log(t)} is a fundamental set of solutions for the homogeneous differential equation, derivatives are computed and the Wronskian is checked. Variation of parameters is then used to find the general solution to the non-homogeneous equation.
Step-by-step explanation:
The question asks to verify whether {1, log(t)} is a fundamental set of solutions for the associated homogeneous differential equation ty" + y' = 0 and to use the method of variation of parameters to find the general solution of the non-homogeneous differential equation ty" + y' = 2/t³ - 3/t⁵.
To verify that {1, log(t)} is a fundamental set of solutions, we show that both functions are solutions to the homogeneous equation. This involves taking their derivatives and plugging them into the equation. Then, we compute the Wronskian, W(t), which is determined by:
W(t) = y₁(t)y₂'(t) - y₂(t)y₁'(t),
where y₁(t) = 1 and y₂(t) = log(t). If the Wronskian is not zero for all t, then the set is fundamental.
Next, using the method of variation of parameters, we find functions u₁(t) and u₂(t) to express the particular solution y(t) as y(t) = u₁(t)y₁(t) + u₂(t)y₂(t). The general solution to the non-homogeneous differential equation will be the sum of the homogeneous solution and this particular solution.