# 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

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.

#### 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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