"Final Answer:
The solution to the initial value problem t²(dx/dt) + 3tx = t⁴ln(t) + 2, with the initial condition x(1) = 0, is given by x(t) = (t⁴/4) - (t³/3) + (2t/3) - (t²/12) - (t/3)ln(t) + t/3.
Step-by-step explanation:
To solve the initial value problem, we utilize the method of integrating factors. The given first-order linear ordinary differential equation is multiplied by an integrating factor, which is t^(-3) in this case, to transform the left side into a total derivative. The resulting equation is then solved by integrating both sides.
Integrating Factor: Multiply the given equation by the integrating factor t^(-3), resulting in t^(-1)(dx/dt) + 3t^(-2)x = tln(t) + 2t^(-3).
Solution by Integration: Integrate both sides of the modified equation. The solution is obtained by integrating each term separately and incorporating the constant of integration. The solution for this problem involves terms with t⁴, t³, t², tln(t), and constants.
Applying Initial Condition: Utilize the initial condition x(1) = 0 to determine the values of the constants in the solution. Substituting t = 1 and x(1) = 0 into the solution allows solving for the constants, resulting in the specific form of the solution provided in the final answer.
The solution incorporates the logarithmic term tln(t) due to the presence of tln(t) on the right side of the original differential equation.