Question

Use Euler's method with step size 0.2 to estimate y(1), where y(x) is the solution of the initial-value problem y' = x2y − 1 2 y2, y(0) = 9. (Round your answer to four decimal places.) y(1) = Need Help?

Answer 1

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, $x_{o} = 0\\y_{0}=9$.

So, if the initial x-value is 0, and the step size is 0.2, the following x-value would be: $x_{1}=0.2$; then $x_{2}=0.4$; $x_{3} =0.6; x_{4} =0.8;x_{5} =1$; and so on.

Now, we have to apply the formula to find each y-value until get the match of $x_{5}=1$, because the problem asks the solution y(1).

According to the Euler's method:

$y_{1} =y_{0} +hF(x_{0};y_{0})\\y_{2} =y_{1} +hF(x_{1};y_{1})\\y_{n} =y_{n-1} +hF(x_{n-1};y_{n-1})$

Where $F(x;y)=x^{2} y-12y^{2}$, and $x_{0} =0; y_{0} =9$; $h=0.2$.

Replacing all values we calculate the y-value assigned to $x_{1}$:

$y_{1} =9+0.2((0)^{2} 9-12(9)^{2})=-185.4$.

Now, $y_{1} =-185.4$, $x_{1} =0.2; h=0.2$. We repeat the process with the new values:

$y_{2} =y_{1} +hF(x_{1};y_{1}) \\y_{2} = -185.4+0.2((0.2)^{2} (-185.4)-12(-185.4)^{2} )\\y_{2}=-82682.47$

Then, we repeat the same process until get the y-value for $x_{5} =1$, which is $y_{5} = -1.0018$, round to four decimal places.

Therefore, $y(1)=-1.0018$.