D3 - Notes for Day : . : First Order Di erentia Equations...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

Page1 / 7

D3 - Notes for Day : . : First Order Di erentia Equations...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online