By Samuel Zaidman

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The problem is to ﬁnd the amount of salt in the tank at any time t. Let Q denote the amount (in pounds) of salt in the tank at any time. The time rate of change of Q, dQ/dt, equals the rate at which salt enters the tank minus the rate at which salt leaves the tank. Salt enters the tank at the rate of be lb/min. To determine the rate at which salt leaves the tank, we ﬁrst calculate the volume of brine in the tank at any time t, which is the initial volume V0 plus the volume of brine added et minus the volume of brine removed ft.

L1 and l2 both real and distinct. 6 can be rewritten as y = k1 cosh l1x + k2 sinh l1x. Case 2. l1 = a + ib, a complex number. 2 must appear in conjugate pairs; thus, the other root is l2 = a − ib. 8) Case 3. l1 = l2. 9) Warning: The above solutions are not valid if the differential equation is not linear or does not have constant coefﬁcients. Consider, for example, the equation y ′′ − x 2 y = 0 . The roots of the characteristic equation are l1 = x and l2 = −x, but the solution is not y = c1e ( x ) x + c2 e ( − x ) x = c1e x + c2 e − x 2 2 Linear equations with variable coefﬁcients are considered in Chapter Twelve.

In other words, they do not depend on y or any derivative of y. 29 Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. 1 is nonhomogeneous. 1 has variable coefﬁcients. 1. 2 has a unique (only one) solution deﬁned throughout Ᏽ. , n − 1) and f(x) = g(x)/bn(x). , n − 1) is continuous on some interval of interest. 7) on a ≤ x ≤ b. 7 is satisﬁed. 7. 7. ,yn(x)} is linearly independent on a ≤ x ≤ b. 2. The nth-order linear homogeneous differential equation L(y) = 0 always has n linearly independent solutions.