use euler's method with delta x=0.1 to estimate y when x=0.4 for the solution curves satisfying y(0)=1 and y(-1)=0.
I made the table but i don't know what values to use or plug in.
x y delta y = dy/dx*delta x
o 1 1*0.1=0.1
To use Euler's method, we approximate the solution to a differential equation by taking small steps and using the slope of the function at each step. In this case, we want to estimate the value of y when x equals 0.4.
First, let's summarize the information we have:
- The step size, Δx, is given as 0.1.
- We have two initial conditions: y(0) = 1 and y(-1) = 0.
Now, let's begin by setting up a table to organize the calculation:
```
x | y
--------------
0 | 1
--------------
```
To calculate the next value of y, we need the slope of the function at x = 0. Using the given initial condition y(0) = 1, we can find the slope at this point. However, since Euler's method requires us to take steps backwards, we first need to find the slope at x = -0.1.
To calculate the slope, we can use the formula:
slope = (y(i) - y(i-1)) / Δx
Substituting the values we have:
slope = (y(0) - y(-0.1)) / Δx
= (1 - y(-0.1)) / 0.1
Since we are given y(-1) = 0, we can estimate y(-0.1) by taking a small step using Euler's method:
```
x | y
--------------
0 | 1
--------------
-0.1 |
--------------
```
Now, we calculate the slope:
slope = (1 - y(-0.1)) / 0.1
To find y(-0.1), we'll use Euler's method:
y(-0.1) = y(0) + slope * Δx
= 1 + slope * 0.1
Having found y(-0.1), we can continue the table:
```
x | y
--------------
0 | 1
--------------
-0.1 | (calculated using Euler's method)
--------------
```
Now, we repeat the process to find the slope at x = -0.2:
slope = (y(-0.1) - y(-0.2)) / Δx
Again, we need to calculate y(-0.2) using Euler's method:
y(-0.2) = y(-0.1) + slope * Δx
We continue this process until we reach x = 0.4:
```
x | y
----------------
0 | 1
----------------
-0.1 | (calculated using Euler's method)
----------------
-0.2 | (calculated using Euler's method)
----------------
-0.3 | (calculated using Euler's method)
----------------
-0.4 | (calculated using Euler's method)
----------------
-0.5 | (calculated using Euler's method)
----------------
-0.6 | (calculated using Euler's method)
----------------
-0.7 | (calculated using Euler's method)
----------------
-0.8 | (calculated using Euler's method)
----------------
-0.9 | (calculated using Euler's method)
----------------
-1 | 0
----------------
```
Finally, once we have y(0.4) calculated in the table, we will have our estimate.