1 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. CHAPTER 25 25.1 (a) The analytical solution can be derived by separation of variables 21.1 dytdty 3ln 1.13tytC Substituting the initial conditions yields C = 0. Taking the exponential gives the final result 31.13ttye The result can be plotted as 012012 (b) Euler’s method with h = 0.5 t y dy/dt 0 1 -1.1 0.5 0.45 -0.3825 1 0.25875 -0.02588 1.5 0.245813 0.282684 2 0.387155 1.122749 Euler’s method with h = 0.25 gives t y dy/dt 0 1 -1.1 0.25 0.725 -0.75219 0.5 0.536953 -0.45641 0.75 0.422851 -0.22728 1 0.36603 -0.0366 1.25 0.356879 0.165057 1.5 0.398143 0.457865 1.75 0.51261 1.005997 2 0.764109 2.215916 The results can be plotted along with the analytical solution as2 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. 012012 (c) The midpoint method with h = 0.5 t y dy/dt tm ym dy/dt-mid 0 1 -1.1 0.25 0.725 -0.75219 0.5 0.623906 -0.53032 0.75 0.491326 -0.26409 1 0.491862 -0.04919 1.25 0.479566 0.221799 1.5 0.602762 0.693176 1.75 0.776056 1.52301 2 1.364267 3.956374 2.25 2.35336 9.32519 with h = 0.25 gives t y dy/dt tm ym dy/dt-mid 0 1 -1.1 0.125 0.8625 -0.93527 0.25 0.766182 -0.79491 0.375 0.666817 -0.63973 0.5 0.60625 -0.51531 0.625 0.541836 -0.38436 0.75 0.510158 -0.27421 0.875 0.475882 -0.15912 1 0.470378 -0.04704 1.125 0.464498 0.076932 1.25 0.489611 0.226445 1.375 0.517916 0.409478 1.5 0.59198 0.680777 1.625 0.677077 1.043122 1.75 0.852761 1.673543 1.875 1.061954 2.565282 2 1.494081 4.332836 2.125 2.035686 6.953139 The results can be plotted along with the analytical solution as 012012 (d) The 4th-order RK method with h = 0.5 gives t y k1 ym k2 ym k3 ye k4 0 1 -1.1000 0.725 -0.7522 0.8120 -0.8424 0.5788 -0.4920 -0.7969 0.5 0.6016 -0.5113 0.4737 -0.2546 0.5379 -0.2891 0.4570 -0.0457 -0.2741 1 0.4645 -0.0465 0.4529 0.2095 0.5169 0.2391 0.5841 0.6717 0.2537 1.5 0.5914 0.6801 0.7614 1.4943 0.9649 1.8937 1.5382 4.4609 1.9861 2 1.5845 4.5949 2.7332 10.8302 4.2920 17.0071 10.0880 51.9532 18.70383 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. 012012 25.2 (a) The analytical solution can be derived by the separation of variables, 14 dyxdxy The integrals can be evaluated to give, 222yxxC Applying the initial condition yields C = 2. Substituting this value and rearranging gives 22222xxy Some selected value can be computed as x y 0 1 0.25 1.410156 0.5 2.25 0.75 3.753906 1 6.25 (b) Euler’s method: (0.25) (0) (0,1)(0,1) (1 4(0)) 1 1(0.25) 1 1(0.25) 1.25yyfhfy (0.5) (0.25) (0.25,1.25)0.25(0.25,1.25) (1 4(0.25)) 1.25 2.236068(0.5) 1.25 2.236068(0.25) 1.809017yy ffy The remaining steps can be implemented and summarized as x y dy/dx 0 1 1 0.25 1.25 2.236068 0.5 1.809017 4.0349914 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. 0.75 2.817765 6.71448 1 4.496385 10.60234 (c) Heun’s method: Predictor: 1(1 4(0)) 1 1k (0.25) 1 1(0.25) 1.25y 2(1 4(0.25)) 1.25 2.236068k Corrector: 1 2.236068(0.25) 1 0.25 1.4045082y The remaining steps can be implemented and summarized as x y k1 xe ye k2dy/dx0 1 1 0.25 1.25 2.236068 1.618034 0.25 1.404508 2.370239 0.5 1.997068 4.23953 3.304885 0.5 2.23073 4.480688 0.75 3.350902 7.322187 5.901438 0.75 3.706089 7.700482 1 5.63121 11.86508 9.782784 1 6.151785 12.4014 1.25 9.252134 18.25039 15.32589 (d) Ralston’s method: Predictor: 1(1 2(0)) 1 1k (0.1875) 1 1(0.1875) 1.1875y 2(1 4(0.1875)) 1.1875 1.907018k Corrector: 1 2(1.907018)(0.25) 1 0.25 1.401173y The remaining steps can be implemented and summarized as x y k1 x + 3/4hy + (3/4)k1hk2dy/dx0 1 1 0.1875 1.1875 1.907018 1.604679 0.25 1.40117 2.36742 0.4375 1.845061 3.735408 3.279412 0.5 2.221023 4.470929 0.6875 3.059322 6.559094 5.863039 0.75 3.686783 7.680398 0.9375 5.126857 10.75522 9.730278 1 6.119352 12.36866 1.1875 8.438476 16.70321 15.25836 (e) RK4 x y k1 ym k2 ym k3 ye k4 0 1 1 1.125 1.59099 1.198874 1.642396 1.410599 2.375373 1.640358 0.25 1.410089 2.374944 1.706957 3.266264 1.818372 3.371176 2.252883 4.502883 3.358785 0.5 2.249786 4.499786 2.812259 5.869427 2.983464 6.045447 3.761147 7.757471 6.014501 0.75 3.753411 7.749489 4.722097 9.778674 4.975745 10.03787 6.262878 12.51287 9.982575 1 6.249054 12.49905 7.811436 15.37192 8.170545 15.72129 10.17938 19.14308 15.638095 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. As on the
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