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Mathematics LibreTexts

Section 3.2 Answers

  • Page ID
    28765
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    1. \(y_{1} = 1.542812500,\: y_{2} = 2.421622101,\: y_{3} = 4.208020541\)

    2. \(y_{1} = 1.220207973,\: y_{2} = 1.489578775.\: y_{3} = 1.819337186\)

    3. \(y_{1} = 1.890687500,\: y_{2} = 1.763784003,\: y_{3} = 1.622698378\)

    4. \(y_{1} = 2.961317914,\: y_{2} = 2.920132727,\: y_{3} = 2.876213748\)

    5. \(y_{1} = 2.478055238,\: y_{2} = 1.844042564,\: y_{3} = 1.313882333\)

    6. 

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact
    \(1.0\) \(56.134480009\) \(55.003390448\) \(54.734674836\) \(54.647937102\)

    7.

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact
    \(2.0\) \(1.353501839\) \(1.353288493\) \(1.353219485\) \(1.353193719\)

    8.

    \(x\) \(h=0.5\) \(h=0.025\) \(h=0.0125\) Exact
    \(1.50\) \(10.141969585\) \(10.396770409\) \(10.472502111\) \(10.500000000\)

     9.

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) \(h=0.1\) \(h=0.05\) \(h=0.025\)
    \(3.0\) \(1.455674816\) \(1.455935127\) \(1.456001289\) \(-0.00818\) \(-0.00207\) \(-0.000518\)
      Approximate Solutions Residuals

    10.

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) \(h=0.1\) \(h=0.05\) \(h=0.025\)
    \(2.0\) \(0.492862999\) \(0.492709931\) \(0.492674855\) \(0.00335\) \(0.000777\) \(0.000187\)
      Approximate Solutions Residuals

    11.

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(1.0\) \(0.660268159\) \(0.660028505\) \(0.659974464\) \(0.659957689\)

    12.

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(2.0\) \(-0.749751364\) \(-0.750637632\) \(-0.750845571\) \(-0.750912371\)

     13. Applying variation of parameters to the given initial value problem \(y = ue^{−3x}\) , where \((A) u' = 1 − 2x, u(0) = 2\). Since \(u''' = 0\), the improved Euler method yields the exact solution of (A). Therefore the improved Euler semilinear method produces the exact solution of the given problem.

    Improved Euler method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact
    \(1.0\) \(0.105660401\) \(0.100924399\) \(0.099893685\) \(0.099574137\)

     

    Improved Euler semilinar method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact
    \(1.0\) \(0.099574137\) \(0.099574137\) \(0.099574137\) \(0.099574137\)

    14. 

    Improved Euler method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(3.0\) \(15.107600968\) \(15.234856000\) \(15.269755072\) \(15.282004826\)

     

    Improved Euler semilinar method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(3.0\) \(15.285231726\) \(15.282812424\) \(15.282206780\) \(15.282004826\)

    15.

    Improved Euler method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(2.0\) \(0.924335375\) \(0.907866081\) \(0.905058201\) \(0.904276722\)

     

    Improved Euler semilinear method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(2.0\) \(0.969670789\) \(0.920861858\) \(0.908438261\) \(0.904276722\)

    16.

    Improved Euler method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(3.0\) \(0.967473721\) \(0.967510790\) \(0.967520062\) \(0.967523153\)

     

    Improved Euler semilinear method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(3.0\) \(0.967473721\) \(0.967510790\) \(0.967520062\) \(0.967523153\)

    17. 

    Improved Euler method
    \(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) "Exact"
    \(1.50\) \(0.349176060\) \(0.345171664\) \(0.344131282\) \(0.343780513\)

     

    Improved Euler semilinear method
    \(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) "Exact"
    \(1.50\) \(0.349350206\) \(0.345216894\) \(0.344142832\) \(0.343780513\)

    18.  

    Improved Euler method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(2.0\) \(0.732679223\) \(0.732721613\) \(0.732667905\) \(0.732638628\)

     

    Improved Euler semilinear method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(2.0\) \(0.732166678\) \(0.732521078\) \(0.732609267\) \(0.732638628\)

    19.  

    Improved Euler method
    \(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) "Exact"
    \(1.50\) \(2.247880315\) \(2.244975181\) \(2.244260143\) \(2.244023982\)

     

    Improved Euler semilinear method
    \(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) "Exact"
    \(1.50\) \(2.248603585\) \(2.245169707\) \(2.244310465\) \(2.244023982\)

    20.

    Improved Euler method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(1.0\) \(0.059071894\) \(0.056999028\) \(0.056553023\) \(0.056415515\)

     

    Improved Euler semilinear method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(1.0\) \(0.056295914\) \(0.056385765\) \(0.056408124\) \(0.056415515\)

    21.

    Improved Euler method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(1.0\) \(50.534556346\) \(53.483947013\) \(54.391544440\) \(54.729594761\)

     

    Improved Euler semilinear method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(1.0\) \(54.709041434\) \(54.724083572\) \(54.728191366\) \(54.729594761\)

    22.

    Improved Euler method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(3.0\) \(1.361395309\) \(1.361379259\) \(1.361382239\) \(1.361383810\)

     

    Improved Euler semilinear method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(3.0\) \(1.375699933\) \(1.364730937\) \(1.362193997\) \(1.361383810\)

    23.

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact
    \(2.0\) \(1.349489056\) \(1.352345900\) \(1.352990822\) \(1.353193719\)

     24.

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact
    \(2.0\) \(1.350890736\) \(1.352667599\) \(1.353067951\) \(1.353193719\)

    25.

    \(x\) \(h=0.05\) \(h=0.025\) \(h=0.0125\) Exact
    \(1.50\) \(10.133021311\) \(10.391655098\) \(10.470731411\) \(10.500000000\)

    26.

    \(x\) \(h=0.05\) \(h=0.025\) \(h=0.0125\) Exact
    \(1.50\) \(10.136329642\) \(10.393419681\) \(10.470731411\) \(10.500000000\)

    27.

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(1.0\) \(0.660846835\) \(0.660189749\) \(0.660016904\) \(0.659957689\)

    28.

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(1.0\) \(0.660658411\) \(0.660136630\) \(0.660002840\) \(0.659957689\)

    29.

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(2.0\) \(-0.750626284\) \(-0.750844513\) \(-0.750895864\) \(-0.751331499\)

    30. 

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(2.0\) \(-0.750335016\) \(-0.750775571\) \(-0.750879100\) \(-0.751331499\)