Respuesta :
Answer:
Euler's method is a numerical method used in calculus to approximate a particular solution of a differential equation. As a numerical method, we have to apply the same procedure many times, until get the desired result.
In first place, we need to know all the values the problem is giving:
- The step size is 0.2; h = 0.2. This step size is a periodical increase of the x-variable, which will allow us to calculate each y-value to each x.
- The problem is asking the solution y(1), which means that we have to find the y-value assigned for x = 1, through the numerical method.
- The initial condition is y(0) = 9. In other words, [tex]x_{o} = 0\\y_{0}=9[/tex].
So, if the initial x-value is 0, and the step size is 0.2, the following x-value would be: [tex]x_{1}=0.2[/tex]; then [tex]x_{2}=0.4[/tex]; [tex]x_{3} =0.6; x_{4} =0.8;x_{5} =1[/tex]; and so on.
Now, we have to apply the formula to find each y-value until get the match of [tex]x_{5}=1[/tex], because the problem asks the solution y(1).
According to the Euler's method:
[tex]y_{1} =y_{0} +hF(x_{0};y_{0})\\y_{2} =y_{1} +hF(x_{1};y_{1})\\y_{n} =y_{n-1} +hF(x_{n-1};y_{n-1})[/tex]
Where [tex]F(x;y)=x^{2} y-12y^{2}[/tex], and [tex]x_{0} =0; y_{0} =9[/tex]; [tex]h=0.2[/tex].
Replacing all values we calculate the y-value assigned to [tex]x_{1}[/tex]:
[tex]y_{1} =9+0.2((0)^{2} 9-12(9)^{2})=-185.4[/tex].
Now, [tex]y_{1} =-185.4[/tex], [tex]x_{1} =0.2; h=0.2[/tex]. We repeat the process with the new values:
[tex]y_{2} =y_{1} +hF(x_{1};y_{1}) \\y_{2} = -185.4+0.2((0.2)^{2} (-185.4)-12(-185.4)^{2} )\\y_{2}=-82682.47[/tex]
Then, we repeat the same process until get the y-value for [tex]x_{5} =1[/tex], which is [tex]y_{5} = -1.0018[/tex], round to four decimal places.
Therefore, [tex]y(1)=-1.0018[/tex].
