Question

solve numerically dy÷dx=2e^x -y at 0.4,0.5 by milne's ,predictor corrector method given their value at the four pointsx=0,0.1,0.2,0.3. y0=2.000,y1=2.010,y2=2.040,y3=2..090​

Answer 1

Concept:

Predictor-corrector algorithms are a subset of algorithms used in numerical analysis to integrate ordinary differential equations, or to locate an unknown function that satisfies a given differential equation.

Given:

dy/dx = 2e^x -y at 0.4,0.5

Find:

By using Milne's ,predictor corrector method find the value of the four points x =0,0.1,0.2,0.3.

Solution:

By using Milne's predictor formula:

$y_{4} = y_{o} + 4h/3[2f1 - f2 + 2f3]$

$y_{4}$ = 2 + 4 x (0.1)/3 [2(0.2003) - 0.4028 + 2(0.6097)]

$y_{4}$ = 2.1623

Now, by using Milne's corrector formula:

$y_{4}$ = $y_{2}$ + h/3 (f2 + 4f3 + f4)

$y_{4} = 2.04 + 0.1/3 [0.4028 + 4(0.6097) + 0.8213]$

$y_{4}$ = 2.1621

$y_{4} = y(x_{4} ) = 2.1621$

Hence, we can say that $y_{4} = y(x_{4} )= y(0.4) = 2.1621$.

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