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Unformatted text preview: Notes for Day : . : First Order Di erentia Equations Previously, weve 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 . In other words, whereever y and y 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 ). Furthermore, no term may contain both y and y . Note that we place no restrictions on how t may appear in the equation. ese di erential equations are linear: y + y- t = y + ( 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 appear as linear terms. ese di erential equations are nonlinear: y - y = t ty + yy = . ( is is nonlinear because y and y 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 y + t = ln t y is linear. So ution: At rst glance, this appears to be nonlinear because y and y appear in the same term. But if we multiply both sides of the equation by y , the equation simpli es to y + 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 . For example, y + y + tan t = is nonhomogeneous, because the term tan t does not contain a y or a y ....
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- Spring '06