SUMMARY
The general solution for the second-order linear differential equation \( t^2y'' - 2y = 0 \) is \( y(t) = C_1t^2 + C_2t^{-1} \), where \( C_1 \) and \( C_2 \) are constants. The solutions \( y_1(t) = t^2 \) and \( y_2(t) = t^{-1} \) are fundamental solutions derived from the characteristic equation. The independent variable is \( t \), not \( x \), which is a common point of confusion in the discussion.
PREREQUISITES
- Understanding of second-order linear differential equations
- Familiarity with the method of solving linear homogeneous equations
- Knowledge of fundamental solutions and their role in constructing general solutions
- Basic calculus, particularly differentiation and integration
NEXT STEPS
- Study the method of undetermined coefficients for solving non-homogeneous differential equations
- Learn about the Wronskian and its application in determining linear independence of solutions
- Explore the theory of differential equations, focusing on the existence and uniqueness theorems
- Practice solving various forms of second-order linear differential equations
USEFUL FOR
Students studying differential equations, mathematicians, and educators looking to deepen their understanding of linear differential equations and their solutions.