Final answer:
The differential equation y' + (1/t)y = 1/t² is solved using an integrating factor, leading to the solution y = (-1/2t) + C/t. If t can take negative values, we must include absolute value signs in the logarithmic part of the solution to correctly apply the domain restrictions of the logarithm.
Step-by-step explanation:
To solve the differential equation y' + (1/t)y = 1/t², we must first identify it as a first-order linear differential equation which can be solved using an integrating factor. The integrating factor, in this case, is e⁰ⁱ⁸ᵐ(ᵉnt t), which simplifies to t⁻¹. Multiplying both sides of the equation by the integrating factor, we have (y/t)' = 1/t³. Integrating both sides gives us y/t = -1/(2t²) + C. Solving for y yields y = (-1/2t) + C/t.
Now, regarding the solution form y = (ln(t) + C)/t, the absolute value signs should be used when applicable due to the domain restrictions of the natural logarithm function, which is undefined for negative values and zero. In the context of this equation, if the variable t represents time or any other quantity that can only take positive values, the absolute value is not necessary. However, if t can take both positive and negative values, the solution would be more correctly written as y = (ln(|t|) + C)/t.