Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: 3. LINEAR EQUATIONS AND BERNOULLI EQUATIONS 11 suppose r(t) = cos2 (t), find the general solution and graph the solution for several different value for the constant C in the same coordinate. Explain the long time behavior of the solutions. Drug Elimination: Let A(t) be the amount of certain drug in the bloodstream, measured by the excess over the natural level of the drug. Then in many situations A(t) will decline at a rate proportional to the current excess amount. That is dA = -A, dt where > 0 is called the elimination parameter of the drug. Suppose for one type of drug, (t) = t21 find the general +4 solution. Discuss its longtime behavior by graphing several solutions in the same coordinate. 3.2. Bernoulli Equations. A Bernoulli equation is an ODE of the form (5) dy + a(x)y = b(x)y n . dx You can see that when n = 1, we get linear equation. So linear equation is a special case of Bernoulli equation. On the other hand if we let v = y 1-n , then dv dy = (1 - n)y -n dx dx n Dividing both side of (5) by y , we have y -n So (6) 1 dv + a(x)v = b(x). 1 - n dx dy + a(x)y 1-n = b(x) dx That is using the transformation v = y 1-n , the Bernoulli equation for y becomes a linear equation for v. Therefore to find the general solution for Bernoulli equation (5) we first find the general solution for the linear equation (6) of v, then using 1 y = v 1-n to find y. ...
View Full Document

This note was uploaded on 10/04/2011 for the course MAP 4231 taught by Professor Dr.han during the Spring '11 term at UNF.

Ask a homework question - tutors are online