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Unformatted text preview: Notes for Day : Â§ . : First Order Di erentia Equations Previously, weâ€™ve classi ed di erential equations based on their order. Chapter focuses exclusively on rstorder di erential equations, and in order to study these, we break the rstorder di erential equations down into ner classi cations: Linear vs. Non inear. A rstorder di erential equation is said to be linear if it is linear in terms of y and y â€² . In other words, whereever y and y â€² appear in the equation, they may not be raised to a power; nor may they be the argument of another function (such as cot y or e y â€² ). Furthermore, no term may contain both y and y â€² . Note that we place no restrictions on how t may appear in the equation. ese di erential equations are linear: â€¢ y â€² + y t = â€¢ y â€² + ( cos t e t ) y = . In this equation, t is used in a nonlinear fashion, but we call the di erential equation linear, because both y and y â€² appear as linear terms. ese di erential equations are nonlinear: â€¢ y â€² y = t â€¢ ty + yy â€² = . ( is is nonlinear because y and y â€² appear in the same term.) A di erential equation may be linear but be written in a form that makes it appear nonlinear, and some algebra is necessary in order to simplify the equation into linear form. Examp e: Show that y â€² y + t = ln t y is linear. So ution: At rst glance, this appears to be nonlinear because y and y â€² appear in the same term. But if we multiply both sides of the equation by y , the equation simpli es to y â€² + t y = ln t , which is linear. Homogeneous vs. Nonhomogeneous A rstorder linear di erential equation is homogeneous if every nonzero term contains either a y or a y â€² . For example, y â€² + y + tan t = is nonhomogeneous, because the term tan t does not contain a y or a y â€² ....
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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
 EDeSturler
 Equations

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