# Differential Equations With Mathematica by Martha L. Abell, James P. Braselton

By Martha L. Abell, James P. Braselton

This quantity thoroughly covers developing, numerically computing and approximating recommendations to bland and partial differential equations. This ebook serves as a hands-on advent to the subject-matter via quite a few examples that designate easy methods to resolve vital purposes utilizing Mathematica.

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**Sample text**

The integral of the left-hand dv + v = -x+2y x -d 2 X x-y side of the equation is found x below with Mathematica. The symbol Log [xl represents the natural logarithm function, Ln(x). 11/10/= Log[-1 + v] _ 3 Log[1 2 + v! 2 Therefore, we have the following equation which must be simplified with logarithmic properties. 1 3 "2Log[-I +v] - "2Log[I +v] Substitution of v = Hence, I-I ( y) x -+1 x 3 Ix = Log[x] +C and simplification yields = Ax2 or 2 x (y - ~) = Ax2 where Aeconsrant. Abell and James P.

C[l)-~. 25 is created in family. Abell and James P. f ... 5 EX, 1. 75 EX, 2. EX} The graphs of the functions in family through the point (0,1). are plotted below. II/76/= -Graphios- To further verify that this initial value problem has a unique solution,DSol ve is used to solve the initial value problem. Abell and James P. , respectively, are perpendicular if the respective slopes satisfy the relationship m = -=!.. Hence, two curves C) and Cz are orthogonal at a point if the m2 respective tangent lines to the curves at that point are perpendicular.

Hence, the relationship is verified, The orthogonalit y 54 DifferentialEquations with Mathematica by Martha L. Abell and James P. Braselton Chapter 3: Applications of First -Order Ordinary Differential Equations 55 is illustrated below with Mathematica by plotting these two curves on a small interval centered at -J2 x=-. 2 First, this particular value of x is entered as a for convenience. ~D OrthogonalTrajectories lo/II= ... 05] to illustrate that they meet at a right angle. 10/2/-... 05}. 2 Find the family of orthogonal trajectories of the family of ellipses t-xy + y2 = c'.