Differential Equations with Mathematica, Third Edition by Martha L. Abell

By Martha L. Abell

The 3rd version of the Differential Equations with Mathematica integrates new purposes from various fields,especially biology, physics, and engineering. the hot instruction manual can be thoroughly appropriate with fresh types of Mathematica and is an ideal advent for Mathematica rookies. The book/CD-ROM package deal includes integrated instructions that we could the consumer clear up difficulties without delay utilizing graphical options. * makes a speciality of the main frequently used good points of Mathematica for the start Mathematica person* CD-ROM comprises all Mathematica inputs from the textual content* New purposes from quite a few fields, together with engineering, biology, and physics* All functions have been accomplished utilizing contemporary models of Mathematica

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Extra resources for Differential Equations with Mathematica, Third Edition

Example text

Entering We use cons to represent the arbitrary constant C to avoid ambiguity with the built-in symbol C. 48 Chapter 2 First-Order Ordinary Differential Equations In[86]:= Clear x, y y x 2 DSolve y x gensol 2y x , x y x ,x Solve ifun Inverse functions are being used by Solve, so some solutions may not be found. 2 C 1 2 Out[86]= y x x x2 finds a general solution of the equation which is equivalent to the one we obtained by hand and names the result gensol. The formula for the solution, which is the second part of the first part of the first part of gensol, is extracted from gensol with gensol[[1,1,2]].

1) See texts like [6], [7], or [3]. x, y a < x < b, c < y < d , 41 42 Chapter 2 First-Order Ordinary Differential Equations containing the point x0 , y0 , there exists an interval x x0 < h centered at x0 on which there exists one and only one solution to the differential equation that satisfies the initial condition. 1); DSolve[y’[x]==f[x,y[x]],y[x],x] f x, y . 1: Solve the initial-value problem dy/ dx y 0 x/ y 0. Does this result contradict the Existence and Uniqueness Theorem? SOLUTION: This equation is solved with DSolve to determine the family of solutions y2 x2 C.

In[21]:= Simplify D fs x, y, t , x, 2 D fs x, y, t , y, 2 1/c0ˆ2 D fs x, y, t , t, 2 //Simplify Out[21]= 0 (b) To graph the normal modes, we choose a b 1. We then use Table and Plot3D to plot Wnm x, y , 0 x 1 and 0 y 1, for n 1, 2, 3, and 4, m 1, 2, and 3. The resulting array of graphics is displayed as a graphics array using Show and GraphicsArray. In Figure 1-3, the first row corresponds to n 1, the second to n 2, and so on; the first column corresponds to m 1, the second to m 3, and so on. In[22]:= tp Table Plot3D Sin n Π x Sin m Π y , x, 0, 1 , y, 0, 1 , DisplayFunction Identity, BoxRatios 1, 1, 1 , PlotPoints 45 , n, 1, 4 , m, 1, 3 Show GraphicsArray tp As indicated in the previous example, without added initial or boundary conditions many differential equations have more than one solution.

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