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By F.M. Dekking, C. Kraaikamp, H.P. Lopuhaä, L.E. Meester
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Extra resources for A Modern Introduction to Differential Equations
For these kinds of equations, we’ll come up with explicit or implicit formulas for their solution curves. This method of solution is often referred to as integrating a differential equation. Next, there is a qualitative way of viewing differential equations. This is a neat geometrical way of studying the behavior of solutions without actually solving the differential equation. The idea is to examine certain pictures or graphs derived from differential equations. Although we can do some of this work by hand, computer algebra systems and many graphing calculators can produce these graphs, and you’ll be expected to use technology when appropriate.
6. In analyzing the change in the percentage of red blood cells in a hospital patient undergoing surgery, the following equation has been used3 H dH , =− dVL EBV where H denotes the hematocrit (percentage of red blood cells in the total volume of blood), VL represents the volume of blood loss, and EBV is the patient’s estimated total blood volume. a. Solve the differential equation for H. 2 A. J. Kroopnick, “Estimating the Cost of National Health Insurance Using Three Simple Models,” Math. and Comp.
In other words, our particle starts out at rest 1 unit (in a positive direction) from the ﬁxed location. We can use these initial conditions to ﬁnd a particular solution of the original differential equation: 1. y(0) = 1 implies that 1 = y(0) = A cos(0) + B sin(0) = A. 2. y (0) = 0 implies that 0 = y (0) = −A sin(0) + B cos(0) = B. Combining the results of (1) and (2), we ﬁnd the particular solution y(t) = cos t. 1 Finding the particular solution of the nth degree equation F t, y, y , y , y , .
A Modern Introduction to Differential Equations by F.M. Dekking, C. Kraaikamp, H.P. Lopuhaä, L.E. Meester