A.3.3: Section 3.3 Answers

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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. $y_{1} = 2.475605264,\: y_{2} = 1.825992433$

$x$	$h=0.1$	$h=0.05$	$h=0.025$	Exact
$1.0$	$54.654509699$	$54.648344019$	$54.647962328$	$54.647937102$

$x$	$h=0.1$	$h=0.05$	$h=0.025$	Exact
$2.0$	$1.353191745$	$1.353193606$	$1.353193712$	$1.353193719$

$x$	$h=0.05$	$h=0.025$	$h=0.0125$	Exact
$1.50$	$10.498658198$	$10.499906266$	$10.499993820$	$10.500000000$

$x$	$h=0.1$	$h=0.05$	$h=0.025$	$h=0.1$	$h=0.05$	$h=0.025$
$3.0$	$1.456023907$	$1.456023403$	$1.456023379$	$0.0000124$	$0.000000611$	$0.0000000333$
	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"
$3.0$	$16.666666988$	$16.666666687$	$16.666666668$	$16.666666667$

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

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

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

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

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

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

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

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