{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# c04 - 18.03 Class 4 First order linear equations...

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

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 x .

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

c04 - 18.03 Class 4 First order linear equations...

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

View Full Document
Ask a homework question - tutors are online