Use Euler's method with a step size of 0.2 to estimate y(1), where y(x) is the solution of the initial value problem y' = 6x+y^2, y(0)=0. Round your final answer to 4 places, but keep more places on the intermediate steps for accuracy.
What, you don't have any work to show?
You can verify your results using the handy calculator at
http://keisan.casio.com/exec/system/1392171850
To use Euler's method to estimate y(1) using a step size of 0.2, we need to follow these steps:
1. Determine the number of iterations required. In this case, the interval from x=0 to x=1 with a step size of 0.2 requires 5 iterations (1/0.2 = 5).
2. Set up a table to keep track of the x and y values at each iteration. The table will have 3 columns: x, y, and y'.
```
| x | y | y' |
|-------|-------|-------|
| 0 | 0 | |
| 0.2 | ... | ... |
| 0.4 | ... | ... |
| 0.6 | ... | ... |
| 0.8 | ... | ... |
| 1.0 | ... | |
```
3. Start by filling in the initial value y(0) = 0.
```
| x | y | y' |
|-------|-------|-------|
| 0 | 0 | |
| 0.2 | ... | ... |
| 0.4 | ... | ... |
| 0.6 | ... | ... |
| 0.8 | ... | ... |
| 1.0 | ... | |
```
4. Fill in the y' values corresponding to each x value. Recall that y' = 6x + y^2.
```
| x | y | y' |
|-------|-------|-------|
| 0 | 0 | 0 |
| 0.2 | ... | ... |
| 0.4 | ... | ... |
| 0.6 | ... | ... |
| 0.8 | ... | ... |
| 1.0 | ... | ... |
```
5. Apply Euler's method to fill in the y values at each iteration. Use the formula: y(i+1) = y(i) + h * y'(i), where h is the step size (0.2).
```
| x | y | y' |
|-------|-------|-------|
| 0 | 0 | 0 |
| 0.2 | ... | ... |
| 0.4 | ... | ... |
| 0.6 | ... | ... |
| 0.8 | ... | ... |
| 1.0 | ... | ... |
```
To calculate the missing values, plug in the known values into the equation y(i+1) = y(i) + h * y'(i).
For example, to find y(0.2), we can use y(0) = 0 and y'(0) = 0:
y(0.2) = y(0) + h * y'(0)
= 0 + 0.2 * 0
= 0
Continue this process for each iteration until you reach x = 1.
6. Once you have populated the entire table, round the final value of y at x = 1 to 4 decimal places.
For instance, if the final value is 1.23456789, round it to 1.2346.
Using this method, you can estimate the value of y(1) using Euler's method with a step size of 0.2 while maintaining accuracy in the intermediate steps.