Using the definition in 35, prove that if r1, r2 and,r3 ,

QUESTION:

Using the definition in Problem 35 , prove that if \(r_{1},\ r_{2}\), and \(r_{3}\) are distinct real numbers, then the functions \(e^{r_{1} t},\ e^{r_{2} t}\), and \(e^{r_{3} t}\) are linearly independent on \((-\infty, \infty)\). [Hint: Assume to the contrary that, say, \(e^{r_{1} t}=c_{1} e^{r_{2} t}+c_{2} e^{r_{3} t}\) for all . Divide by \(e^{r_{2}t}\) to get \(e^{\left(r_{1}-r_{2}\right) t}=c_{1}+c_{2} e^{\left(r_{3}-r_{2}\right) t}\) and then differentiate to deduce that \(e^{\left(r_{1}-r_{2}\right) t}\) and \(e^{\left(r_{3}-r_{2}\right) t}\) are linearly dependent, which is a contradiction. (Why?)]

Equation Transcription:

Text Transcription:

r_{1}, r_{2}

r_{3}

e^{r_{1}t}, e^{r_{2}t}

e^{r_{3}t}

(-infinity, infinity)

e^{r_{1} t}=c_{1} e^{r_{2} t}+c_{2} e^{r_{3} t}

e^{r_{2}t}

e^{(r_{1}-r_{2})t}=c_{1}+c_{2} e^{(r_{3}-r_{2})t}

e^{(r_{1}-r_{2})t}

e^{(r_{3}-r_{2})t}