# 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 suggestions to bland and partial differential equations. This publication serves as a hands-on creation to the subject-matter via various examples that designate the way to remedy very important purposes utilizing Mathematica.

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Abell and James P. ll{6/= , -x + 2 r[x) DSolve[y [x] == 2 x - y [x I ' y(x], xl However, it can be solved by making the substitution v = I. Note that by using the chain rule, we have x dy -d X dv + v . By substi h d'f~terentia. I equation .. mvo I'vmg v: = x -d sntution, we h ave tel X is separable. This leads to 2 - v dv v2 -1 = dx . 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).

18 Solve the initial value problem dy dx _!. y yeO) = O. Does this result contradict the Existence and Uniqueness Theorem? Solution: This equation can be solved by separation of variables which yields the equation ydy = x dx which is easily solved to determine the family of solutions y - x2 = C. Hence, if x = Y = 0, then C = O. Abell and James P. Braselton 51 Chapter 2: First-Order Ordinary Differential Equations Therefore, two solutions pass through (0,0), y Mathematica below. = x and y = -x. 1 The solutions are extracted from the output list of ivp for convenience with the following commands.

Abell and James P. Braselton Chapter 3: Applications of First -Order Ordinary Differential Equations 56 Solution: We first determine the differential equation satisfied by the family of ellipses. Implicit differentiation yields ~ (X2_xy+y2=C2) or 2x-y-x dy + 2y dy =0. :'l= . 2 x - y[x] ({y [x] -) -(-x + 2 y[x])}) J] 1IIIil Iliill ] I This formula is extracted with the command sol [ [1, 1, 2] ] . lt/4/; Therefore, the family of orthogonal trajectories satisfies dy = -x + 2y . dx 2x-y This is a homogeneous first-order equation which cannot be solved directly withDSolve as indicated below.