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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 rst-order di erential equations, and in order to study these, we break the rst-order di erential equations down into ner classi cations: Linear vs. Non inear. A rst-order 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 non-linear 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 rst-order 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