ECUACIONES DIFERENCIALES

EJEMPLO

CALCULAR EL VALOR DE y = ??x = 2

Condicion inicial X0=Y0 =

PROCESOPASO h=(2-1)/n

EULERi xi yi0 1 21 1.1 2.2083333332 1.2 2.3787206443 1.3 2.5206370044 1.4 2.6403845815 1.5 2.7424475196 1.6 2.8301575097 1.7 2.9060648028 1.8 2.972164199 1.9 3.03004186910 2 3.080975658

HEUNi xi yi0 1 21 1.1 2.2083692622 1.2 2.3861916973 1.3 2.5386453874 1.4 2.669913195 1.5 2.7834179566 1.6 2.8819769777 1.7 2.9679165938 1.8 3.0431624829 1.9 3.10931255310 2 3.167696269

�� =((1+/�� 〖 )2� 〗^2)/((2 ^2) ^2))+� (�∗�

ECUACIONES DIFERENCIALES

12

PROCESO0.1

EULERye=yi +0.10*(1+y2)/(1+x2)(xy)

2.20833333332.37872064392.5206370042.6403845812.74244751912.83015750932.90606480222.97216418973.03004186943.0809756581

HEUNye=yi +0.10*(1+2y2)/(2+x2)(xy^2) Yh=

2.2083333333 2.20836932.3787555491 2.38619172.5279537486 2.53864542.6581119123 2.66991322.7716170008 2.7834182.8707302286 2.8819772.9574749765 2.96791663.033612167 3.04316253.100651919 3.10931263.1598791531 3.1676963

�� =((1+/�� 〖 )2� 〗^2)/((2 ^2) ^2))+� (�∗�

0.9 1.1 1.3 1.5 1.7 1.9 2.11.9

2.1

2.3

2.5

2.7

2.9

3.1

3.3

GRAFICO

X

Y

0.9 1.1 1.3 1.5 1.7 1.9 2.11.9

2.1

2.3

2.5

2.7

2.9

3.1

3.3

GRAFICO

X

Y

0.9 1.1 1.3 1.5 1.7 1.9 2.11.9

2.1

2.3

2.5

2.7

2.9

3.1

3.3

GRAFICO

X

Y

Ejemplo

CALCULAR EL VALOR DE y =??4

SI

SOLUCIONPROCESO

PASO h=(2-1)/n

EULERi xi yi0 1 21 1.05 2.33752 1.1 2.73715687043 1.15 3.22004537424 1.2 3.81639062275 1.25 4.57133258076 1.3 5.55531775777 1.35 6.88405055258 1.4 8.75970599879 1.45 11.564024442310 1.5 16.093761340711 1.55 24.250941554412 1.6 41.517043039713 1.65 88.794044350114 1.7 291.187674938615 1.75 2331.15548533416 1.8 125009.942573717 1.85 331496632.997618 1.9 2.1908721E+01519 1.95 9.0063240E+02820 2 1.4337564E+056

HEUNi xi yi0 1 21 1.05 2.00721861952 1.1 2.0141142687

�� =3/ (/�� (� 〖 2�〗^2 ^3))+�

3 1.15 2.02071369474 1.2 2.02704036195 1.25 2.03311498456 1.3 2.0389559537 1.35 2.04457968048 1.4 2.05000088519 1.45 2.05523282410 1.5 2.060287485711 1.55 2.065175752212 1.6 2.069907534913 1.65 2.0744918914 1.7 2.078937116415 1.75 2.083250839516 1.8 2.087440083917 1.85 2.091511335918 1.9 2.095470597819 1.95 2.099323435220 2 2.1030750192

condicion inicialx0= 1y0= 2

0.05

EULERye=yi +0.05*((1+2y^2)^2)/((2+x^2)*(x*y^2))

2.33752.73715687043.22004537423.81639062274.57133258075.55531775776.88405055258.759705998711.564024442316.093761340724.250941554441.517043039788.7940443501291.18767493862331.1554853344125009.94257369331496632.9976472190872053451490

900632396021175000000000000001.43375638310249E+056

HEUNye=yi +0.05*((1+2y^2)^2)/((2+x^2)*(x*y^2)) Yh=

2.3375 2.00721861952.319776256 2.01411426872.3040198703 2.0207136947

0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.10

20000000000000000000000000000000000000000000000000000000

40000000000000000000000000000000000000000000000000000000

60000000000000000000000000000000000000000000000000000000

80000000000000000000000000000000000000000000000000000000

100000000000000000000000000000000000000000000000000000000

120000000000000000000000000000000000000000000000000000000

140000000000000000000000000000000000000000000000000000000

160000000000000000000000000000000000000000000000000000000

GRAFICO

x

Y

2.2899980375 2.02704036192.2775120806 2.03311498452.2663910388 2.0389559532.2564867415 2.04457968042.2476699714 2.05000088512.2398274355 2.0552328242.2328593451 2.06028748572.2266774568 2.06517575222.2212034699 2.06990753492.2163677004 2.074491892.2121079737 2.07893711642.208368691 2.08325083952.2051000359 2.08744008392.2022572952 2.09151133592.1998002743 2.09547059782.1976927894 2.09932343522.1959022268 2.1030750192

CALCULAR EL VALOR DE y = ??x = 2

Condicion inicial X0= 1Y0 = 2

0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.10

20000000000000000000000000000000000000000000000000000000

40000000000000000000000000000000000000000000000000000000

60000000000000000000000000000000000000000000000000000000

80000000000000000000000000000000000000000000000000000000

100000000000000000000000000000000000000000000000000000000

120000000000000000000000000000000000000000000000000000000

140000000000000000000000000000000000000000000000000000000

160000000000000000000000000000000000000000000000000000000

GRAFICO

x

Y

�� =((1+/�� 〖 )2� 〗^2)/((2 ^2) ^2))+� (�∗�

2.18 2.2 2.22 2.24 2.26 2.28 2.3 2.32 2.34 2.361.94

1.96

1.98

2

2.02

2.04

2.06

2.08

2.1

2.12

f(x) = 3.424166366x̂ 2 - 16.1437621224x + 21.037101094R² = 0.9942983415

GRAFICO

x

y

2.18 2.2 2.22 2.24 2.26 2.28 2.3 2.32 2.34 2.361.94

1.96

1.98

2

2.02

2.04

2.06

2.08

2.1

2.12

f(x) = 3.424166366x̂ 2 - 16.1437621224x + 21.037101094R² = 0.9942983415

GRAFICO

x

y

0.05

0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.10

20000000000000000000000000000000000000000000000000000000

40000000000000000000000000000000000000000000000000000000

60000000000000000000000000000000000000000000000000000000

80000000000000000000000000000000000000000000000000000000

100000000000000000000000000000000000000000000000000000000

120000000000000000000000000000000000000000000000000000000

140000000000000000000000000000000000000000000000000000000

160000000000000000000000000000000000000000000000000000000

GRAFICO

x

Y

�� =((1+/�� 〖 )2� 〗^2)/((2 ^2) ^2))+� (�∗�

2.18 2.2 2.22 2.24 2.26 2.28 2.3 2.32 2.34 2.361.94

1.96

1.98

2

2.02

2.04

2.06

2.08

2.1

2.12

f(x) = 3.424166366x̂ 2 - 16.1437621224x + 21.037101094R² = 0.9942983415

GRAFICO

x

y

2.18 2.2 2.22 2.24 2.26 2.28 2.3 2.32 2.34 2.361.94

1.96

1.98

2

2.02

2.04

2.06

2.08

2.1

2.12

f(x) = 3.424166366x̂ 2 - 16.1437621224x + 21.037101094R² = 0.9942983415

GRAFICO

x

y

Ejemplo

CALCULAR EL VALOR DE y =?? si x = 1

SI

SOLUCIONPROCESO

PASO h=(2-0)/n

EULERi xi yi0 0 21 0.2 2.22 0.4 2.415983 0.6 2.639926064 0.8 2.864097235 1 3.081322246 1.2 3.285267817 1.4 3.470667858 1.6 3.633504829 1.8 3.77113613

10 2 3.8823611

HEUNi xi yi0 0 21 0.2 2.207992 0.4 2.427953033 0.6 2.652011654 0.8 2.872709735 1 3.083295026 1.2 3.277967837 1.4 3.452086348 1.6 3.602320489 1.8 3.72674861

10 2 3.82489351

�� =sen /�� 〖 〖〖〖〖〖〖〖〖〖〖〖〖 /2 +���� 〗

condicion inicial Columna1

0

2

0.2

EULERye=yi +0.2*(senx/2)+cosx

2.22.41597999892.63992606392.86409722823.08132223853.28526780743.4706678533.633504819

3.77113613273.8823610957

HEUNye=yi +0.2*(senx/2)+cosx Yh=

2.2 2.20798999942.5051306929 2.42795303142.7537021904 2.6520116462.9192527231 2.87270973333.001962465 3.0832950229

3.0282896573 3.27796783023.0419849984 3.4520863363.0893267134 3.60232047593.2034285989 3.72674861423.3928440334 3.8248935105

x0 =

y0 =

0 0.5 1 1.5 2 2.50.7

1.2

1.7

2.2

2.7

3.2

3.7

4.2

GRAFICA

X

Y

0 0.5 1 1.5 2 2.50.7

1.2

1.7

2.2

2.7

3.2

3.7

4.2

GRAFICA

X

Y

DIFERENCIAS DIVIDIDAS

EJEMPLO

SE TIENE LAS SIGUIENTE TABLA

PRESION Columna1

3/8" 2"30 8.57 16.5845 10.24 20.3665 11.67 23.8776 12.94 27.5985 14.64 30.3299 17.1 36.6

110 18 39HALLAR f(75)

SOLUCION

PRESION Ф 1º

3/8" D30 8.5745 10.24 0.1113365 11.67 0.0715076 12.94 0.1154585 14.64 0.1888999 17.1 0.17571

110 18 0.0818181818

x = 75 f(x) 12.7869130053

PRESION Ф 1º

2" D30 16.5845 20.36 0.2520065 23.87 0.1755076 27.59 0.3381885 30.32 0.3033399 36.6 0.44857

110 39 0.21818

x = 75 f(x) 27.2603031952

CAUDAL pie3/seg

lbs/pulg2

lbs/pulg2

pie3/seg

lbs/pulg2

pie3/seg

SOLUCION

2º 3º 4º 5º 6º

D D D D D

-0.001140.00142 0.00005560.00367 0.0000563 0.000000014-0.00057 -0.0001248 -0.000003355 -0.00000004883290

-0.00375584 -0.0000936 0.000000694 0.00000006229319 1.389076E-009

2º 3º 4º 5º 6º

D D D D D

-0.002190.00525 0.0001616-0.00174 -0.0001748 -0.0000061160.00631 0.0002370 0.000007625 0.00000019913261-0.00922 -0.0004568 -0.000015417 -0.00000035448052 -6.92016E-009

1 2 3 4 5 6 70

20

40

60

80

100

120

f(x) = 13.1428571429x + 20.2857142857R² = 0.9880924586

f(x) = 1.6064285714x + 6.8828571429R² = 0.9918882073

f(x) = 3.7925x + 12.59R² = 0.9920565996

GRAFICA

lbs/pulg2 Linear (lbs/pulg2) 3/8"

Linear (3/8") 2" Linear (2")

1 2 3 4 5 6 70

20

40

60

80

100

120

f(x) = 13.1428571429x + 20.2857142857R² = 0.9880924586

f(x) = 1.6064285714x + 6.8828571429R² = 0.9918882073

f(x) = 3.7925x + 12.59R² = 0.9920565996

GRAFICA

lbs/pulg2 Linear (lbs/pulg2) 3/8"

Linear (3/8") 2" Linear (2")