chapter2.page11

chapter2.page11 - 3. LINEAR EQUATIONS AND BERNOULLI...

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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. ...
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