mgR2 (y + R)2" 0, y"(O) =vo. in Java Implementation barcode pdf417 in Java mgR2 (y + R)2" 0, y"(O) =vo.

mgR2 (y + R)2" 0, y"(O) =vo. generate, create pdf417 none with java projects ASP.NET We rescale the dependent and independent variables, using y = ax, t = (3T,. where x is t Java PDF417 he new dependent variable, T is the new independent variable, and a and (3 are scaling constants. Note that we want a to have the same. 4.1. INTRODUCTION units as y a PDF 417 for Java nd f3 to have the same units as t, so that the new variables will be dimensionless. By the chain rule,. ! (!~). ~ (d aXdT) dT dT dT d2 x dt dt f32 dT 2 Substituting into our initial value problem, we have mad2x ---f32 PDF 417 for Java dT2 x(O) mgR2 (ax + RF" adx 0, dT (0) = va,. d 2x dT2 x(O) f32gR2 a(ax + R)2". = f3 v o. dx (0) dT There are se veral ways to choose a and f3 to make the problem dimensionless. We take the following approach: First elimate the parameters in the initial conditions by making a = f3vo. Our problem is now.

x(O). The choice f 3 = vol g gives f3 the units of time, as desired, and results in the dimensionless equation. Now if the s PDF-417 2d barcode for Java quare of the initial velocity of the rocket is small compared to the product of the radius of the earth and the acceleration of gravity, then the initial value problem contains a small parameter. 4. PERTURBATION METHODS The result o j2se PDF-417 2d barcode f our calculation is a dimensionless problem with a small parameter:. d2x dT2 x(O). 1 (Ex+1)2". (4.1) 1. (4.

2). ~~ (0) =. Note that th e process of making the problem dimensionless has reduced the number of parameters from four to one. 6. Now that we have a problem with a small parameter, we can compute a perturbation series that will provide much useful information about the solution when E is small.

. Example 4.4 We substitute a perturbation series X(T, E) = XO(T) + EXI(T) + E2X2(T) + ... into equations (4.1), (4.2) and get d2xo ~XI 2d2 x2 - d + E 2 +E - d + ...

2 2 dT T T Xo(O) + EXI(O) + E2X2(0) + ...

dxo dXI 2dx2 dT (0) + E dT (0) + E dT (0) + ...

. In order to j2se PDF417 equate coefficients of like powers of E in the differential equation, we must first change the right-hand side into a series in powers of E. We can do this quickly by recalling the binomial series: ( 1 + z)n = 1 + nz which converges for equation becomes. 2! < 1. With z = EXo + n(n -. 1) Z2 + .. _.

the differential + E2xI + ---,. --+E--+E --+--dT2 dT2 dT2 d2xo d2x2 -[1+(-2)(EXO+E2Xl+---). (-3)(-2). (lOXO + E Xl + ---) + -- -jE Equating the terms on each side of each equation that do not involve gives us the problem for Xo:. Xo(O). ~o (0) = 1.. The solution tomcat PDF417 can be found by direct integration:. Xo =T -. 4.1. INTRODUCTION Next, we equate the coefficients of E on each side of each equation to obtain Xl(O). 2xo = 2 (T _T;) ,. ~l (0) = 0. and then integrate to find the solution = - --.. 3 12. T3 T4 Similarly, the problem for d2 x2 with solution given by X2(T) =. ~: (0) = 0,. -4 + 60 E 360 . ~ T~ T (E)),. For brevity, let"s include only terms through in the perturbation series:. T- T; + E (T; _ ~~) + 0 (E2). 0) (0. where T(E) i awt PDF-417 2d barcode s the time when the rocket returns to the earth"s surface. Note that the next term in the series actually involves the expressions E 2 T 4 , E2 T 5 , and E2T6, but the dependence on T can be supressed here since T varies over a bounded interval..

In the original variables, the perturbation series is vot _. ~gt2 + ~ [~ 2 g2 R 3 (gt)3 _ (gt)4] Vo 12 Vo + ...

.. The expressi jboss PDF417 on vot - gt 2/2 represents the position of the rocket if we assume that the force of gravity is constant throughout the flight. The next segment of the perturbation series is positive and represents the first-order correction, revealing (as expected) that the rocket actually flies higher due to the fact that the force of gravity diminishes with increasing altitude. 6 We saw in Example 4.

1 that a perturbation series does not necessarily yield a valid approximation of the solution, so our discussion for Example 4.4 is incomplete. The most commonly used method of checking an approximation of this type is to generate a numerical approximation for certain initial and parameter values and test it against the analytic approximation.

Of course, this is only a partial check since we can"t try all possible values. Here is a theorem that can be used to verify approximations for problems of the type discussed previously..

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