c04 - 18.03 Class 4, Feb 15, 2006 First order linear...

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18.03 Class 4, Feb 15, 2006 First order linear equations: solutions. [1] form" Definition: A "linear ODE" is one that can be put in the "standard _____________________________ | | | x' + p(t)x = q(t) | (*) |_____________________________| On Monday we looked at the Homogeneous case, q(t) = 0 : x' + p(t) x = 0 . This is separable, and the solution is xh = C e^{- integral p(t) dt} Now for the general case. Example: x' + kT = kT_ext, the (heat) diffusion equation. k is the "coupling constant." Let's take it to be 1/3. (This cooler cost $16.95 at Target.) Suppose the temperature outside is rising at a constant rate: say T_ext = 60 + 6t (in hours after 10:00) and we need an initial condition: x(0) = 32 . So the equation is x' + (1/3) x = 20 + 2t , x(0) = 32 . This isn't separable: it's something new. We'll describe a method which works for ANY first order liner ODE. [2] Method: "variation of parameter," or "trial solution": (1) First solve the "associated homogeneous equation" x' + p(t) x = 0 (*)_h Write xh for a nonzero solution to it. (2) Then make the substitution x = xh u , and solve for u . (3) Finally, don't forget to substitute back in to get
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This note was uploaded on 01/31/2011 for the course MAT 17A taught by Professor Staff during the Winter '08 term at UC Davis.

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c04 - 18.03 Class 4, Feb 15, 2006 First order linear...

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