# Thread: "Match the third order linear equations with their fundamental solution sets"

1. ## "Match the third order linear equations with their fundamental solution sets"

The question is attached. How do I go about doing this? If someone does just one of them that should be enough for me to get it.

Judging via y&#39;&#39;&#39; - 5y&#39;&#39; &#43; 6y&#39;&#61; 0 - Wolfram|Alpha, the answer for #1 is C since the general solution has those 3 particular solutions but how do I do it without cheating?

Any help would be greatly appreciated!

2. ## Re: "Match the third order linear equations with their fundamental solution sets"

The key to all of those problems, except # 5, is to write the characteristic equation, which you obtain simply by assuming a solution of the form

$\displaystyle y(t)=e^{mt},$

and plug it into the equation. Derivatives become corresponding powers of $\displaystyle m,$ and the exponentials always cancel because they're nonzero. You must then solve the resulting polynomial. Complex solutions you can resolve into trigonometric functions via the formula

$\displaystyle e^{it}=\cos(t)+i\sin(t).$