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Using Modified Eulers Method ,Find an Approximate Value of Y at X = 0.2 in Two Step Taking H=0.1 and Using Three Iteration Given that D Y D X = X + 3 Y , Y = 1 When X = 0. - Applied Mathematics 2

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Sum

Using Modified Eulers method ,find an approximate value of y At x = 0.2 in two step taking h=0.1 and using three iteration Given that `(dy)/(dx)=x+3y` , y = 1 when x = 0.

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Solution

(I) `(dy)/(dx)=x+3y`  𝒙𝟎=𝟎, 𝒚𝟎=𝟏, 𝒉=𝟎.𝟏

`y_1^0=y_0+hf(x_0,y_0)=1+0.1(3)=1.3`

`y_1^(n+1)=y_0+h/2[f(x_0,y_0)+f(x_1,y_1^n)]`

Iteration x1 `y_1^n` `x_1y_1^n` `y_1^(n+1)`
0 0.1 1.3 4 1.35
1 0.1 1.35 4.15 1.3575
2 0.1 1.3575 4.1725 1.3587

y(0.1)=1.3587

(II) 𝒙𝟏=𝟎.𝟏,𝒚𝟏=𝟏.𝟑𝟓𝟖𝟕

`y_2^0=1.77631`

`y_2^(n+1)=y_1+h/2[f(x_2,y_2)+f(x_2,y_2^n)]`

Iteration x2 `y_1^n` `x_2y_2^n` `y_2^(n+1)`
0 0.2 1.77631 5.52893 1.8439
1 0.2 1.8439 5.7317 1.8540
2 0.2 1.8540 5.762 1.8556

y(0.2)=1.8556

Concept: Modified Euler Method
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