Final answer:
The student's question involves a second-order linear non-homogeneous differential equation in mathematics, typically encountered in college. It requires finding the function y(t) that satisfies the equation, which is a concept applied in ordinary differential equations.
Step-by-step explanation:
The differential equation provided, t²y′′ + ty′ - y = t², is a mathematical problem that deals with finding the function y(t) that satisfies the equation. This type of equation is seen in topics like ordinary differential equations or applied mathematics, often at the college level.
To approach solving this equation, one would likely classify it as a second-order linear non-homogeneous differential equation. It can sometimes be solved using methods such as variation of parameters or by finding a particular solution that satisfies the non-homogeneous part and the complementary solution for the homogeneous part.
The trajectory equation examples given in the question are illustrating principles of kinematics, which is a subfield of physics. However, these don't directly relate to solving the initial differential equation. A common method in solving such differential equations involves identifying known initial conditions, finding a general solution to the associated homogeneous equation, and then obtaining a particular solution that fits the non-homogeneous part.