
Existence and Uniqueness
Hi everyone,
I'm having trouble with the ideas involved with the uniquesness theorem for a linear nth order IVP.
As an example:
The initial value problem
$\displaystyle 3y'''+5y''y'+7y=0,$ $\displaystyle y(1)=0$, $\displaystyle y'(1)=0,$ $\displaystyle y''(1)=0$.
Now, I can see that $\displaystyle y=0$ is a trivial solution here, but by the theorem, this has got to be the ONLY solution on any interval containing 1. I don't see this. Is this true? Is there no other solution to this problem ?

Let's suppose that the solution od the DE around $\displaystyle x=1$ is of the form...
$\displaystyle \displaystyle y(x)= \sum_{n=0}^{\infty} y^{(n)} (1)\ \frac{(x1)^{n}}{n!}$ (1)
... and our scope is to find the $\displaystyle y^{(n)}(1)$ for all n. The 'initial conditions' give us $\displaystyle y(1)=y^{'}(1)=y^{''}(1)=0$. The sucessive drivatives are...
$\displaystyle \displaystyle y^{(3)} (1)=  \frac{5}{3}\ y^{(2)} (1) + \frac{1}{3}\ y^{(1)} (1)  \frac{7}{3}\ y(1)=0$ (2)
$\displaystyle \displaystyle y^{(4)} (1) =  \frac{5}{3}\ y^{(3)} (1) + \frac{1}{3}\ y^{(2)} (1)  \frac{7}{3}\ y^{(1)} (1) =0 $ (3)
... and so one. All the derivatives in (1) vanish so that y=0 is the only solution...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$