Answer:
[tex]y_2=0.512[/tex]
Step-by-step explanation:
We are given that
Step size=0.2
Initial value problem
y'=2-3xy
f(x,y)=2-3xy
y(1)=0
We have to find the value of [tex]y_2[/tex]
Euler's method of approximation
[tex]y_{n+1)=y_n+hf(x_n,y_n)[/tex]
We have [tex]y_0=0,x_0=1[/tex]
Substitute the value then we get when n=1
[tex]y_1=y_0+hf(x_0,y_0)[/tex]
[tex]y_1=0+(0.2)(2-3(1)(0))[/tex]
[tex]y_1=0.4[/tex]
[tex]x_1=x_0+h=1+0.2=1.2[/tex]
[tex]y_2=y_1+hf(x_1,y_1)[/tex]
[tex]y_2=0.4+0.2(2-3(1.2)(0.4))[/tex]
[tex]y_2=0.4+0.2\times 0.56[/tex]
[tex]y_2=0.4+0.112=0.512[/tex]
[tex]y_2=0.512[/tex]
The initial value of [tex]y' = 2 - 3\cdot x \cdot y[/tex] for [tex]x = 2[/tex] is [tex]y_{5} \approx 0.3668[/tex].
The Euler's method is a multistage numerical method, to estimate a point of a given solution based on a given initial value. The expressions required in this method are presented below:
[tex]f(x_{i}, y_{i}) = \frac{dy}{dx}|_{P_{i}}[/tex] (1)
[tex]x_{i+1} = x_{i} + h[/tex] (2)
[tex]y_{i+1} = y_{i} + h\cdot f(x_{i}, y_{i})[/tex] (3)
Where [tex]h[/tex] is the step size.
Now we start iterating this differential equation:
[tex]x_{o} = 1[/tex]
[tex]y_{o} = 0[/tex]
By (1):
[tex]f(x_{o}, y_{o}) = 2[/tex]
By (2):
[tex]x_{1} = 1.2[/tex]
By (3):
[tex]y_{1} = 0.4[/tex]
[tex]x_{1} = 1.2[/tex]
[tex]y_{1} = 0.4[/tex]
By (1):
[tex]f(x_{1}, y_{1}) = 0.56[/tex]
By (2):
[tex]x_{2} = 1.4[/tex]
By (3):
[tex]y_{2} = 0.512[/tex]
[tex]x_{2} = 1.4[/tex]
[tex]y_{2} = 0.512[/tex]
By (1):
[tex]f(x_{2}, y_{2}) = -0.150[/tex]
By (2):
[tex]x_{3} = 1.6[/tex]
By (3):
[tex]y_{3} = 0.41[/tex]
[tex]x_{3} = 1.6[/tex]
[tex]y_{3} = 0.41[/tex]
By (1):
[tex]f(x_{3}, y_{3}) = 0.032[/tex]
By (2):
[tex]x_{4} = 1.8[/tex]
By (3):
[tex]y_{4} = 0.416[/tex]
[tex]x_{4} = 1.8[/tex]
[tex]y_{4} = 0.416[/tex]
By (1):
[tex]f(x_{4}, y_{4}) = -0.246[/tex]
By (2):
[tex]x_{5} = 2[/tex]
By (3):
[tex]y_{5} = 0.3668[/tex]
The initial value of [tex]y' = 2 - 3\cdot x \cdot y[/tex] for [tex]x = 2[/tex] is [tex]y_{5} \approx 0.3668[/tex]. [tex]\blacksquare[/tex]
To learn more on Euler's method, we kindly invite to check this verified question: https://brainly.com/question/15237275
