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Unformatted text preview: Notes for Day : . : Introduction to Second Order Linear Di erentia Equations When studying rst-order linear homogeneous equations, we o en had solutions of the form y = e kt for some constant k . Is it possible that the same solutions might work in second-order linear homogeneous equations as well? e answer is: sometimes . In section . , we cover the types of second-order linear homogeneous equations for which y = e kt still works as a solution. Seeing why y = e kt works as a solution will give us some insight into other second-order equations as well. Examp e: Lets begin with the di erential equation ay + by + cy = (this is not in standard form). If y = e kt is a solution to the equation, then we get: a ( k e kt )+ b ( ke kt )+ c ( e kt ) = Factoring out the common term e kt , we get: e kt ( ak + bk + c ) = Since e kt can never equal zero, were le with ak + bk + c = , but this is just a quadratic equation, and we know how to factor a quadratic equation to solve for...
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This note was uploaded on 04/05/2008 for the course MATH 2214 taught by Professor Edesturler during the Spring '06 term at Virginia Tech.
- Spring '06