n 10 1 2 3 4 5 dx Yn 7 = 3/3; y Let f(x, y) = x²/y. We let x = 0.1 and yo = 7 and pick a step size h = 0.2. Euler's method is the the following algorithm. From x, and yn, our approximations to the solution of the differential equation at the nth stage, we find the next stage by computing Xn+1 = xn + h, Yn+1 = yn + h. f(xn, yn). Complete the following table. Your answers should be accurate to at least seven decimal places. Xn 0.1 y(0.1) = 7. The exact solution can also be found using separation of variables. It is y(x) = Thus the actual value of the function at the point x = 1.1 y(1.1) =
n 10 1 2 3 4 5 dx Yn 7 = 3/3; y Let f(x, y) = x²/y. We let x = 0.1 and yo = 7 and pick a step size h = 0.2. Euler's method is the the following algorithm. From x, and yn, our approximations to the solution of the differential equation at the nth stage, we find the next stage by computing Xn+1 = xn + h, Yn+1 = yn + h. f(xn, yn). Complete the following table. Your answers should be accurate to at least seven decimal places. Xn 0.1 y(0.1) = 7. The exact solution can also be found using separation of variables. It is y(x) = Thus the actual value of the function at the point x = 1.1 y(1.1) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Differential Equation Example Using Euler’s Method**
We are given the differential equation:
\[
\frac{dy}{dx} = \frac{x^2}{y}
\]
with the initial condition \( y(0.1) = 7 \).
Let \( f(x, y) = \frac{x^2}{y} \).
We use the initial point \( x_0 = 0.1 \) and \( y_0 = 7 \), and choose a step size \( h = 0.2 \). Using Euler’s method, we apply the following algorithm to approximate the solution of the differential equation at each stage \( n \). We calculate the next stage by computing:
\[
x_{n+1} = x_n + h
\]
\[
y_{n+1} = y_n + h \cdot f(x_n, y_n)
\]
Complete the following table. Your answers should be accurate to at least seven decimal places.
| \( n \) | \( x_n \) | \( y_n \) |
|---------|----------|-----------|
| 0 | 0.1 | 7 |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
The exact solution can also be found using separation of variables. It is:
\[ y(x) = \]
Thus, the actual value of the function at the point \( x = 1.1 \) is:
\[ y(1.1) = \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1f1c68a1-c113-41cc-b0cf-42f2666e5687%2F261e3e17-e72f-429a-a394-6ed2382bf732%2Fo9xw1u_processed.png&w=3840&q=75)
Transcribed Image Text:**Differential Equation Example Using Euler’s Method**
We are given the differential equation:
\[
\frac{dy}{dx} = \frac{x^2}{y}
\]
with the initial condition \( y(0.1) = 7 \).
Let \( f(x, y) = \frac{x^2}{y} \).
We use the initial point \( x_0 = 0.1 \) and \( y_0 = 7 \), and choose a step size \( h = 0.2 \). Using Euler’s method, we apply the following algorithm to approximate the solution of the differential equation at each stage \( n \). We calculate the next stage by computing:
\[
x_{n+1} = x_n + h
\]
\[
y_{n+1} = y_n + h \cdot f(x_n, y_n)
\]
Complete the following table. Your answers should be accurate to at least seven decimal places.
| \( n \) | \( x_n \) | \( y_n \) |
|---------|----------|-----------|
| 0 | 0.1 | 7 |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
The exact solution can also be found using separation of variables. It is:
\[ y(x) = \]
Thus, the actual value of the function at the point \( x = 1.1 \) is:
\[ y(1.1) = \]
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