# Existence and Uniqueness Theorem for Linear Equations

• Sep 30th 2011, 11:41 AM
alexmahone
Existence and Uniqueness Theorem for Linear Equations
Prove that an nth-order homogenous linear differential equation satisfying the hypotheses of the Existence and Uniqueness Theorem for Linear Equations has n linearly independent solutions \$\displaystyle y_1,\ y_2,\ ...,\ y_n\$.

(Suggestion: Let \$\displaystyle y_i\$ be the unique solution such that

\$\displaystyle y_i^{(i-1)}(a)=1\$ and \$\displaystyle y_i^{(k)}(a)=0\$ if \$\displaystyle k\neq i-1\$.)
• Oct 1st 2011, 01:47 AM
alexmahone
Re: Existence and Uniqueness Theorem for Linear Equations
Assume \$\displaystyle c_1y_1+c_2y_2+...+c_ny_n=0\$.

Evaluating at \$\displaystyle a\$,

\$\displaystyle c_1y_1(a)+c_2y_2(a)+...+c_ny_n(a)=0\$

\$\displaystyle c_1=0\$

Differentiating once,

\$\displaystyle c_1y_1'+c_2y_2'+...+c_ny_n'=0\$

Evaluating at \$\displaystyle a\$,

\$\displaystyle c_1y_1'(a)+c_2y_2'(a)+...+c_ny_n'(a)=0\$

\$\displaystyle c_2=0\$

etc.

Thus, \$\displaystyle c_1=c_2=c_3=...=c_n=0\$.

So, \$\displaystyle y_1,\ y_2,\ ...,\ y_n\$ are linearly independent.

Is this right?