By Martha L. Abell, James P. Braselton

This quantity thoroughly covers developing, numerically computing and approximating recommendations to bland and partial differential equations. This booklet serves as a hands-on advent to the subject-matter via a number of examples that specify how one can clear up vital purposes utilizing Mathematica.

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Yo),then there exists an interval Ix- xo h centered at Xoon which there exists one and only one solution to the differential equation which satisfies the initial condition. 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.

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. 211 Outf71/= -Sqrt[x2) Finally, the two solutions of the initial value problem are plotted.

Dx 2x-y This is a homogeneous first-order equation which cannot be solved directly withDSolve as indicated below. First, the right-hand side of the differential equation is defined as orth below for use in the next step. Then, we attempt to solve the equation with DSolve. Unfortunately, DSolve is unable to solve the equation. 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 .

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