Thread: Family of Solutions

Given y = c_1 + c_2*t^2 is a two-parameter family of solutions of ty’’ – y’ = 0 on the interval (-inf, inf), show that c_1 and c_2 (both constants) cannot be found so that a member of the family satisfies the following initialy conditions y(0) = 0, y'(0) = 1.

Why doesn't this violate the Existence of a Unique Solution theorem (Let a_n(t), a_(n-1)(T), ..., a_(1)(t), a_(0)(t) and g(t) be continuous on an interval I and let a_n(t) not equal 0 for every t in this interval. If t = t_0 is any point in the interval, then a sol'n of the IVP (non-homogeneous eq.) exists on the interval and is unique.

Originally Posted by Ideasman

Given y = c_1 + c_2*t^2 is a two-parameter family of solutions of ty’’ – y’ = 0 on the interval (-inf, inf), show that c_1 and c_2 (both constants) cannot be found so that a member of the family satisfies the following initialy conditions y(0) = 0, y'(0) = 1.

Why doesn't this violate the Existence of a Unique Solution theorem (Let a_n(t), a_(n-1)(T), ..., a_(1)(t), a_(0)(t) and g(t) be continuous on an interval I and let a_n(t) not equal 0 for every t in this interval. If t = t_0 is any point in the interval, then a sol'n of the IVP (non-homogeneous eq.) exists on the interval and is unique.

Because this (fundamental) theorem only works when,

This is the form we need.
Divide by and we get which is not continous on any open interval containg the origin!