# Thread: Existence and Uniqueness Theorem for Linear Equations

1. ## 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 $\displaystyle k\neq i-1$.)

2. ## 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?