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Mathematics LibreTexts

Section 3.3 Answers

  • Page ID
    28900
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    1. \(y_{1}=1.550598190, \: y_{2} = 2.469649729\)

    2. \(y_{1} = 1.221551366,\: y_{2} = 1.492920208\)

    3. \(y_{1} = 1.890339767,\: y_{2} = 1.763094323\)

    4. \(y_{1} = 2.961316248,\: y_{2} = 2.920128958\)

    5. \(y_{1} = 2.475605264,\: y_{2} = 1.825992433\)

    6. 

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact
    \(1.0\) \(54.654509699\) \(54.648344019\) \(54.647962328\) \(54.647937102\)

    7.

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact
    \(2.0\) \(1.353191745\) \(1.353193606\) \(1.353193712\) \(1.353193719\)

     8.

    \(x\) \(h=0.05\) \(h=0.025\) \(h=0.0125\) Exact
    \(1.50\) \(10.498658198\) \(10.499906266\) \(10.499993820\) \(10.500000000\)

    9.

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) \(h=0.1\) \(h=0.05\) \(h=0.025\)
    \(3.0\) \(1.456023907\) \(1.456023403\) \(1.456023379\) \(0.0000124\) \(0.000000611\) \(0.0000000333\)
      Approximate Solutions Residuals

     10.

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) \(h=0.1\) \(h=0.05\) \(h=0.025\)
    \(2.0\) \(0.492663789\) \(0.492663738\) \(0.492663736\) \(0.000000902\) \(0.0000000508\) \(0.00000000302\)
      Approximate Solutions Residuals

    11.

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(1.0\) \(0.659957046\) \(0.659957646\) \(0.659957686\) \(0.659957689\)

     12.

    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(2.0\) \(-0.750911103\) \(-0.750912294\) \(-0.750912367\) \(-0.750912371\)

    13.  Applying variation of parameters to the given initial value problem yields \(y = ue^{−3x}\), where \((A) u' = 1 − 4x + 3x^{2} − 4x^{3} , u(0) = −3\). Since \(u^{(5)} = 0\), the Runge-Kutta method yields the exact solution of (A). Therefore the Euler semilinear method produces the exact solution of the given problem.

    clipboard_e9e428930ffce428ad2d7df8d39937a25.png

    clipboard_e075f364cb890bde142bc6294dc1ba24e.png

    14.

    Runge-Kutta method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(3.0\) \(15.281660036\) \(15.281981407\) \(15.282003300\) \(15.282004826\)

     

    Runge-Kutta semilinear method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(3.0\) \(15.282005990\) \(15.282004899\) \(15.282004831\) \(15.282004826\)

    15.

    Runge-Kutta method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(2.0\) \(0.904678156\) \(0.904295772\) \(0.904277759\) \(0.904276722\)

     

    Runge-Kutta semilinear method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(2.0\) \(0.904592215\) \(0.904297062\) \(0.904278004\) \(0.904276722\)

    16.

    Runge-Kutta method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(3.0\) \(0.967523147\) \(0.967523152\) \(0.967523153\) \(0.967523153\)

     

    Runge-Kutta semilinear method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(3.0\) \(0.967523147\) \(0.967523152\) \(0.967523153\) \(0.967523153\)

    17.

    Runge-Kutta method
    \(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) "Exact"
    \(1.50\) \(0.343839158\) \(0.343784814\) \(0.343780796\) \(0.343780513\)

     clipboard_e7cd069c28f0e49fea809ca150442f812.png

    18. 

    Runge-Kutta method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(2.0\) \(0.732633229\) \(0.732638318\) \(0.732638609\) \(0.732638628\)

     

    Runge-Kutta semilinear method
    \(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) "Exact"
    \(2.0\) \(0.732639212\) \(0.732638663\) \(0.732638630\) \(0.732638628\)

    19.

    Runge-Kutta method
    \(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) "Exact"
    \(1.50\) \(2.244025683\) \(2.244024088\) \(2.244023989\) \(2.244023982\)

     

    Runge-Kutta semilinear method
    \(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) "Exact"
    \(1.50\) \(2.244025081\) \(2.244024051\) \(2.244023987\) \(2.244023982\)

    20.

    Runge-Kutta method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(1.0\) \(0.056426886\) \(0.056416137\) \(0.056415552\) \(0.056415515\)

     

    Runge-Kutta semilinear method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(1.0\) \(0.056415185\) \(0.056415495\) \(0.056415514\) \(0.056415515\)

    21.  

    Runge-Kutta method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(1.0\) \(54.695901186\) \(54.727111858\) \(54.729426250\) \(54.729594761\)

     

    Runge-Kutta semilinear method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(1.0\) \(54.729099966\) \(54.729561720\) \(54.729592658\) \(54.729594761\)

    22.

    Runge-Kutta method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(3.0\) \(1.361384082\) \(1.361383812\) \(1.361383809\) \(1.361383810\)

     

    Runge-Kutta semilinear method
    \(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) "Exact"
    \(3.0\) \(1.361456502\) \(1.361388196\) \(1.361384079\) \(1.361383810\)

    24.

    \(x\) \(h=.1\) \(h=.05\) \(h=.025\) Exact
    \(2.00\) \(-1.000000000\) \(-1.000000000\) \(-1.000000000\) \(-1.000000000\)

    25.

    \(x\) \(h=.1\) \(h=.05\) \(h=.025\) "Exact"
    \(1.00\) \(1.000000000\) \(1.000000000\) \(1.000000000\) \(1.000000000\)

     26.

    \(x\) \(h=.1\) \(h=.05\) \(h=.025\) "Exact"
    \(1.50\) \(4.142171279\) \(4.142170553\) \(4.142170508\) \(4.142170505\)

    27.

    \(x\) \(h=.1\) \(h=.05\) \(h=.025\) "Exact"
    \(3.0\) \(16.666666988\) \(16.666666687\) \(16.666666668\) \(16.666666667\)