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Mathematics LibreTexts

Section 3.1 Answers

  • Page ID
    28707
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    1. \(y_{1}= 1.450000000,\: y_{2} = 2.085625000,\: y_{3} = 3.079099746\)

    2. \(y_{1} = 1.200000000,\: y_{2} = 1.440415946,\: y_{3} = 1.729880994\)

    3. \(y_{1} = 1.900000000,\: y_{2} = 1.781375000,\: y_{3} = 1.646612970\)

    4. \(y_{1} = 2.962500000,\: y_{2} = 2.922635828,\: y_{3} = 2.880205639\)

    5. \(y_{1} = 2.513274123,\: y_{2} = 1.814517822,\: y_{3} = 1.216364496\)

    6. 

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact
    \(1.0\) \(48.298147362\) \(51.492825643\) \(53.076673685\) \(54.647937102\)

    7.

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact
    \(2.0\) \(1.390242009\) \(1.370996758\) \(1.361921132\) \(1.353193719\)

    8.

    \(x\) \(h=0.05\) \(h=0.025\) \(h=0.0125\) Exact
    \(1.50\) \(7.886170437\) \(8.852463793\) \(9.548039907\) \(10.500000000\)

    9. 

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) \(h=0.1\) \(h=0.05\) \(h=0.025\)
    \(3.0\) \(1.469458241\) \(1.462514486\) \(1.459217010\) \(0.3210\) \(0.1537\) \(0.0753\)
      Approximate Solutions Residuals

    10. 

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) \(h=0.1\) \(h=0.05\) \(h=0.025\)
    \(2.0\) \(0.473456737\) \(0.483227470\) \(0.487986391\) \(-0.3129\) \(-0.1563\) \(-0.0781\)
      Approximate Solutions Residuals

    11. 

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(1.0\) \(0.691066797\) \(0.676269516\) \(0.668327471\) \(0.659957689\)

    12. 

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(2.0\) \(-0.772381768\) \(-0.761510960\) \(-0.756179726\) \(-0.750912371\)

    13. 

    Euler's Method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact
    \(1.0\) \(0.538871178\) \(0.593002325\) \(0.620131525\) \(0.647231889\)

     

    Euler semilinear Method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact
    \(1.0\) \(0.647231889\) \(0.647231889\) \(0.647231889\) \(0.647231889\)

    Applying variation of parameters to the given initial value problem yields \(y = ue^{−3x}\) , where (A) \(u' = 7, u(0) = 6\). Since \(u''= 0\), Euler’s method yields the exact solution of (A). Therefore the Euler semilinear method produces the exact solution of the given problem

    14. 

    Euler's Method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(3.0\) \(12.804226135\) \(13.912944662\) \(14.559623055\) \(15.282004826\)

     

    Euler semilinear method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(3.0\) \(15.354122287\) \(15.317257705\) \(15.299429421\) \(15.282004826\)

    15.

    Euler's method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(2.0\) \(0.867565004\) \(0.885719263\) \(0.895024772\) \(0.904276722\)

     

    Euler's semilinear method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(2.0\) \(0.569670789\) \(0.720861858\) \(0.808438261\) \(0.904276722\)

    16.

    Euler's method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(3.0\) \(0.922094379\) \(0.945604800\) \(0.956752868\) \(0.967523153\)

     

    Euler semilinear method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(3.0\) \(0.993954754\) \(0.980751307\) \(0.974140320\) \(0.967523153\)

    17. 

    Euler's method
    \(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) "Exact"
    \(1.50\) \(0.319892131\) \(0.330797109\) \(0.337020123\) \(0.343780513\)

     

    Euler semilinear method
    \(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) "Exact"
    \(1.50\) \(0.305596953\) \(0.323340268\) \(0.333204519\) \(0.343780513\)

    18.

    Euler's method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(2.0\) \(0.754572560\) \(0.743869878\) \(0.738303914\) \(0.732638628\)

     

    Euler semilinear method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(2.0\) \(0.722610454\) \(0.727742966\) \(0.730220211\) \(0.732638628\)

    19.

    Euler's method
    \(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) "Exact"
    \(1.50\) \(2.175959970\) \(2.210259554\) \(2.227207500\) \(2.244023982\)

     

    Euler semilinear method
    \(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) "Exact"
    \(1.50\) \(2.117953342\) \(2.179844585\) \(2.211647904\) \(2.244023982\)

    20. 

    Euler's method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(1.0\) \(0.032105117\) \(0.043997045\) \(0.050159310\) \(0.056415515\)

     

    Euler's semilinear method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(1.0\) \(0.056020154\) \(0.056243980\) \(0.056336491\) \(0.056415515\)

    21.

    Euler's method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(1.0\) \(28.987816656\) \(38.426957516\) \(45.367269688\) \(54.729594761\)

     

    Euler's semilinar method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(1.0\) \(54.709134946\) \(54.724150485\) \(54.728228015\) \(54.729594761\)

    22.

    Euler's method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(3.0\) \(1.361427907\) \(1.361320824\) \(1.361332589\) \(1.361383810\)

     

    Euler's semilinar method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(3.0\) \(1.291345518\) \(1.326535737\) \(1.344004102\) \(1.361383810\)