math34 finals coritico
DESCRIPTION
math34TRANSCRIPT
Submitted By: Coritico,Marc Angelo C.
Taylor Series
F(x,y) = dy/dx = y' = 2x^2y^2
y(0) = 1
F = y' = 2x^2 y^2
F' = y'' = 2x^2 y^2[4x^2y] + 4xy^2 = 8x^4 y^3 + 4xy^2
F'' = y''' = 2x^2 y^2[24x^4 y^2 + 8xy] + [32x^3 y^3 + 4y^2] = 48x^6 y^4 + 16x^3 y^3 + 32x^3 y^3 + 4y^2 = 48x^6 y^4 + 48x^3 y^3 + 4y^2
i h xi yi yi+10 0.1 0 1 1.0006671 0.1 0.1 1.000667 1.0053522 0.1 0.2 1.005352 1.0182853 0.1 0.3 1.018285 1.0444384 0.1 0.4 1.044438 1.0905885 0.1 0.5 1.090588 1.1674616 0.1 0.6 1.167461 1.2945517 0.1 0.7 1.294551 1.5127978 0.1 0.8 1.512797 1.9257629 0.1 0.9 1.925762 2.883328
10 0.1 1 2.883328 6.421382
0 ≤ x ≤ 1
0 0.2 0.4 0.6 0.8 1 1.20
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3.5
Taylor Series
0 0.2 0.4 0.6 0.8 1 1.20
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Modified Euler
Submitted By: Coritico,Marc Angelo C.
Modified Euler Runge Kutta - Second Order
F(x,y) = dy/dx = y' = 2x^2y^2 y' = 2x^2y^2 = F(xi,yi)
y(0) = 1 k1 = hF(xi,yi) = h(2x^2y^2) = 2hx^2y^2
i h xi yi yi+1 k2 = hF(xi+h,yi+k1) = h[2(x^2+h)(y^2+2hx^2y^2)]0 0.1 0 1 11 0.1 0.1 1 1.002 i2 0.1 0.2 1.002 1.010032 03 0.1 0.3 1.010032 1.028395 14 0.1 0.4 1.028395 1.062238 25 0.1 0.5 1.062238 1.118656 36 0.1 0.6 1.118656 1.208756 47 0.1 0.7 1.208756 1.351943 58 0.1 0.8 1.351943 1.585894 69 0.1 0.9 1.585894 1.993334 7
10 0.1 1 1.993334 2.78801 89
10
0 ≤ x ≤ 1
0 0.2 0.4 0.6 0.8 1 1.20
0.5
1
1.5
2
2.5
Modified Euler
0 0.2 0.4 0.6 0.8 1 1.20
0.5
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Runge Kutta -Second Order
Runge Kutta - Second Order Runge Kutta - 3rd Order
y' = 2x^2y^2 = F(xi,yi) y' = 2x^2y^2 = F(xi,yi)
k1 = hF(xi,yi) = h(2x^2y^2) = 2hx^2y^2 k1 = 2hx^2y^2
k2 = hF(xi+h,yi+k1) = h[2(x^2+h)(y^2+2hx^2y^2)] k2 = h2(x^2 + h/2)(y^2 + k1/2)
h xi yi k1 k2 yi+1 k3 = h2(x^2 + h)(y^2 - k1 +2k2)0.1 0 1 0 0.02 1.010.1 0.1 1.01 0.00204 0.022487 1.0222640.1 0.2 1.022264 0.00836 0.029495 1.041191 i0.1 0.3 1.041191 0.019513 0.041937 1.071916 00.1 0.4 1.071916 0.036768 0.06166 1.12113 10.1 0.5 1.12113 0.062847 0.092385 1.198746 20.1 0.6 1.198746 0.103463 0.141722 1.321338 30.1 0.7 1.321338 0.171102 0.22621 1.519994 40.1 0.8 1.519994 0.295729 0.385705 1.860711 50.1 0.9 1.860711 0.560884 0.73221 2.507258 60.1 1 2.507258 1.257269 1.659595 3.96569 7
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0 0.2 0.4 0.6 0.8 1 1.20
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Runge Kutta -Second Order
0 0.2 0.4 0.6 0.8 1 1.20
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Runge Kutta - 3rd Order
Runge Kutta - 3rd Order
y' = 2x^2y^2 = F(xi,yi) yi+1 = yi + (1/6)(k1 + 4k2 + k3)
k1 = 2hx^2y^2
k2 = h2(x^2 + h/2)(y^2 + k1/2)
k3 = h2(x^2 + h)(y^2 - k1 +2k2)
h xi yi k1 k2 k3 yi+10.1 0 1 0 0.01 0.0204 1.0100670.1 0.1 1.010067 0.00204 0.012255 0.022939 1.02240.1 0.2 1.0224 0.008362 0.018891 0.030092 1.0414030.1 0.3 1.041403 0.019521 0.03064 0.042799 1.0722160.1 0.4 1.072216 0.036789 0.049058 0.062971 1.1215480.1 0.5 1.121548 0.062893 0.077359 0.094479 1.1993490.1 0.6 1.199349 0.103568 0.122198 0.145293 1.3222910.1 0.7 1.322291 0.171349 0.198086 0.232847 1.5217150.1 0.8 1.521715 0.296399 0.340006 0.399486 1.8643660.1 0.9 1.864366 0.56309 0.646274 0.765368 2.5166250.1 1 2.516625 1.266681 1.463016 1.758406 3.99615
0 0.2 0.4 0.6 0.8 1 1.20
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Runge Kutta - 3rd Order
Runge Kutta - 4th Order
y' = 2x^2y^2 = F(xi,yi) yi+1 = yi + (1/6)(k1 + 2k2 + 2k3 + k4)
k1 = 2hx^2y^2 k3 = h2(x^2 + h)(y^2 - k1 +2k2)
k2 = h2(x^2 + h/2)(y^2 + k1/2) k4 = h2(x^2 + h)(y^2 + k3)
i h xi yi k1 k2 k3 k4 yi+10 0.1 0 1 0 0.01 0.0204 0.020408 1.0135351 0.1 0.1 1.013535 0.002055 0.012339 0.023097 0.023108 1.0295412 0.1 0.2 1.029541 0.00848 0.019155 0.030514 0.030533 1.0525993 0.1 0.3 1.052599 0.019943 0.031302 0.043724 0.043764 1.0882264 0.1 0.4 1.088226 0.037896 0.050534 0.064865 0.064953 1.1438345 0.1 0.5 1.143834 0.065418 0.080464 0.098271 0.098464 1.2307256 0.1 0.6 1.230725 0.109057 0.128675 0.152994 0.153426 1.3683627 0.1 0.7 1.368362 0.183497 0.21213 0.249355 0.250369 1.5945028 0.1 0.8 1.594502 0.325432 0.373311 0.438617 0.441196 1.9929159 0.1 0.9 1.992915 0.643417 0.738468 0.874552 0.88202 2.784828
10 0.1 1 2.784828 1.551054 1.791467 2.153173 2.179857 4.721527
0 0.2 0.4 0.6 0.8 1 1.20
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Runge Kutta - 4th Order