ecuaciones.xlsx
TRANSCRIPT
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
2 2.2 2.4 2.6 2.8 3 3.2 3.40
0.5
1
1.5
2
2.5
3
3.5
GRAFICA
x
y
2 2.2 2.4 2.6 2.8 3 3.2 3.40
0.5
1
1.5
2
2.5
3
3.5
GRAFICA
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
2 2.2 2.4 2.6 2.8 3 3.2 3.40
0.5
1
1.5
2
2.5
3
3.5
GRAFICA
x
y
2 2.2 2.4 2.6 2.8 3 3.2 3.40
0.5
1
1.5
2
2.5
3
3.5
GRAFICA
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 PROCESO
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
PROCESO0.2
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
2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.60
0.5
1
1.5
2
2.5
3
3.5
4
4.5
GRAFICO
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
2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.60
0.5
1
1.5
2
2.5
3
3.5
4
4.5
GRAFICO
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")