Family of Solutions

• Sep 27th 2007, 08:35 AM
Ideasman
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.
• Sep 29th 2007, 04:02 PM
ThePerfectHacker
Quote:

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,
$\displaystyle y''+f(t)y'+g(t)y=0$
This is the form we need.
Divide by $\displaystyle t$ and we get $\displaystyle f(t) = \frac{1}{t}$ which is not continous on any open interval containg the origin!