Answer:
Step-by-step explanation:
Given is a differential equation
[tex]y' = y − 3x, y(4) = 2[/tex]
Here we have [tex]x_0 = 4 and y_0 =2[/tex]
To find y1, y2, y3, y4
Step size = 0.5
[tex]x_1=4.5, x_2 =5,...[/tex]
y_1 = y_0 +0.5f(x_0,y_0)
Here f(x,y) = y-3x
Applying this successively we get
y1 = 2+0.5(2-12) = -3
y2 = -3+0.5(-3-13.5) = -11.25
y3 =-11.25+0.5(-11.25-15) =-24.375
y4=-24.375+0.5(-24.375-16.5)=-44.8125
The approximate values for [tex]y_{1}[/tex], [tex]y_{2}[/tex], [tex]y_{3}[/tex] and [tex]y_{4}[/tex] are 6, 9.75, 12.375 and 14.438, respectively.
In this question we must analyze a first order differential equation by numerical methods.
The Euler's method is a multistep numerical method to estimate values of first order differential equations, whose formulae are described below:
[tex]x_{i+1} = x_{i}+h[/tex] (1)
[tex]y_{i+1} = y_{i} + h\cdot \frac{dy}{dx}|_{(x_{i}, y_{i})}[/tex] (2)
Where [tex]h[/tex] is the step size.
If we know that [tex]x_{o} = 4[/tex] and [tex]y_{o} = 2[/tex], then the values of [tex]y[/tex] are, respectively:
[tex]x_{o} = 4[/tex], [tex]x_{1} = 4.5[/tex], [tex]y_{1} = 6[/tex]
[tex]x_{1} = 4.5[/tex], [tex]x_{2} = 5[/tex], [tex]y_{2} = 9.75[/tex]
[tex]x_{2} = 5[/tex], [tex]x_{3} = 5.5[/tex], [tex]y_{3} = 12.375[/tex]
[tex]x_{3} = 5.5[/tex], [tex]x_{4} = 6.0[/tex], [tex]y_{4} = 14.438[/tex]
The approximate values for [tex]y_{1}[/tex], [tex]y_{2}[/tex], [tex]y_{3}[/tex] and [tex]y_{4}[/tex] are 6, 9.75, 12.375 and 14.438, respectively. [tex]\blacksquare[/tex]
To learn more on Euler's method, we kindly invite to check this verified question: https://brainly.com/question/11325462
