# Existence and Uniqueness

• Feb 11th 2011, 11:35 AM
VonNemo19
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 ?
• Feb 11th 2011, 12:17 PM
chisigma
Let's suppose that the solution od the DE around $\displaystyle x=1$ is of the form...

$\displaystyle \displaystyle y(x)= \sum_{n=0}^{\infty} y^{(n)} (1)\ \frac{(x-1)^{n}}{n!}$ (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

$\displaystyle \chi$ $\displaystyle \sigma$