Second-Order Linear ODE (5) - Existence and Uniqueness of Solution

9. Existence and Uniqueness of Solution

 

$ y''+py'+qy=0 $

$ y(x_{0})=K_{0}, y'(x_{0})=K_{1} $

 

Theorem 1) Existence and Uniqueness Theorem for Initial Value Problems

p, q: continuous on I ($ x_{0}\in I $)

Then $ y''+py'+qy=0 $ ($ y(x_{0})=K_{0}, y'(x_{0})=K_{1} $) has a unique solution on I.

 

Theorem 2) Linear Dependence and Independence of Solutions

i) $ y_{1}, y_{2} $: Linearly Dependent $ \Leftrightarrow W=W(y_{1}, y_{2})=y_{1}y_{2}'-y_{1}'y_{2}=0 $ for some $ x_{0}\in I $

ii) If W=0 at $ x=x_{0}\in I \Rightarrow  $ W=0 for all $ x\in I $

 

Theorem 3) Existence of a General Solution

p, q: continuous on I

Then $ y''+py'+qy=0 $ has a general solution on I.

 

Theorem 4) A General Solution Includes All Solutions

p, q: continuous on I

Then every solution $ y=Y(x)=C_{1}y_{1}(x)+C_{2}y_{2}(x) $

$ y_{1}, y_{2}$: any basis of solutions

Hence $ y''+py'+qy=0 $ does not have singular solutions