Given that y c 1 c 2 x 2 is a two parameter family of solutions of xy ' ' y ' 0 on the interval (, ), show that constants c 1 and c 2 cannot be found so that a member of the family satisfies the initial conditions y(0) 0, y '(0) 1 Explain why this does not violate Theorem 4 1 1 ...

From y c 1 c 2 x 2 we find y 2c ...

Given that y = c 1 + c 2 x 2 is a two-parameter family of solutions

Question:

Given that y = c₁ + c₂x² is a two-parameter family of solutions of xy'' – y' = 0 on the interval (‑∞, ∞), show that constants c₁ and c₂ cannot be found so that a member of the family satisfies the initial conditions y(0) = 0, y'(0) = 1. Explain why this does not violate Theorem 4.1.1.