Solve x^2+y =dy/dx , y(0)=1 ,h=0.01,0.05,x(0)=0, using eulars method 

To solve the first-order differential equation \( \frac{dy}{dx} = x^2 + y \) using Euler's method with initial conditions \( y(0) = 1 \), step sizes \( h = 0.01 \) and \( h = 0.05 \), and \( x(0) = 0 \), we proceed as follows:

### Problem Setup:
The differential equation is:

\[
\frac{dy}{dx} = x^2 + y
\]

With the initial condition \( y(0) = 1 \), we need to apply Euler's method for two different step sizes \( h = 0.01 \) and \( h = 0.05 \).

### Euler's Method:
Euler's method is a numerical technique for solving differential equations. Given the differential equation \( \frac{dy}{dx} = f(x, y) \) and initial values \( x_0 \) and \( y_0 \), the formula for the next point is:

\[
y_{n+1} = y_n + h \cdot f(x_n, y_n)
\]
\[
x_{n+1} = x_n + h
\]

Where \( f(x_n, y_n) = x_n^2 + y_n \).

### Step-by-Step Approach:
1. Start at \( x = 0 \) with \( y(0) = 1 \).
2. Apply the Euler's formula to compute the next values of \( y \) and \( x \) for each step.

Now, I will compute the values for each step size. Let's begin with \( h = 0.01 \) and \( h = 0.05 \). I'll calculate the values numerically for the first few steps.

#### For \( h = 0.01 \):

1. At \( x_0 = 0 \), \( y_0 = 1 \):
   \[
   y_1 = 1 + 0.01 \cdot (0^2 + 1) = 1.01
   \]
   \[
   x_1 = 0 + 0.01 = 0.01
   \]
2. At \( x_1 = 0.01 \), \( y_1 = 1.01 \):
   \[
   y_2 = 1.01 + 0.01 \cdot (0.01^2 + 1.01) = 1.01 + 0.01 \cdot (0.0001 + 1.01) = 1.010 + 0.0101 = 1.0201
   \]
   \[
   x_2 = 0.01 + 0.01 = 0.02
   \]

#### For \( h = 0.05 \):

1. At \( x_0 = 0 \), \( y_0 = 1 \):
   \[
   y_1 = 1 + 0.05 \cdot (0^2 + 1) = 1.05
   \]
   \[
   x_1 = 0 + 0.05 = 0.05
   \]
2. At \( x_1 = 0.05 \), \( y_1 = 1.05 \):
   \[
   y_2 = 1.05 + 0.05 \cdot (0.05^2 + 1.05) = 1.05 + 0.05 \cdot (0.0025 + 1.05) = 1.05 + 0.05 \cdot 1.0525 = 1.05 + 0.052625 = 1.102625
   \]
   \[
   x_2 = 0.05 + 0.05 = 0.10
   \]

I can continue calculating more steps or generate tables for these computations. Would you like to see a complete table for both step sizes \( h = 0.01 \) and \( h = 0.05 \) up to a specific value of \( x \)? Or would you like further details on any part?

### Follow-Up Questions:
1. Would you like to see the results for a larger interval of \( x \)?
2. Should we use a different numerical method like Runge-Kutta for comparison?
3. Do you need help visualizing these results on a graph?
4. Should we investigate the effect of using smaller or larger step sizes?
5. Would you like an explanation of how Euler’s method accuracy is affected by \( h \)?

**Tip:** Euler's method is easy to implement but can be inaccurate for stiff equations or when larger step sizes are used. Smaller step sizes yield more accurate results.

Solve x^2 + y = dy/dx, y(0) = 1, h = 0.01, 0.05, x(0) = 0, using Euler's method

Learn how to solve the first-order differential equation dy/dx = x^2 + y using Euler's method with initial condition y(0) = 1. This guide covers step sizes h = 0.01 and h = 0.05, providing a detailed step-by-step solution and analysis of the numerical method's accuracy.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation