Numerical Solutions of Ordinary Differential Equations Using Euler's Method

Introduction

• Focus on finding numerical solutions of Ordinary Differential Equations (ODEs) using Euler's Method.
• Importance in solving differential equations where traditional methods are difficult or impossible to apply.

Euler's Method

• Simple and widely used numerical technique for approximating solutions of ODEs.
• Based on the idea of using the slope at one point to estimate the value at the next point.

Differential Equation Example

• Given ODE: ( y' = f(x, y) )
• Initial condition: ( y(0) = y_0 )
• Calculate subsequent values using: ( y_{n+1} = y_n + h * f(x_n, y_n) )
  • Where:
    • ( y_n ) is the current value.
    • ( h ) is the step size.
    • ( f(x_n, y_n) ) is the function evaluation at ( x_n, y_n ).

Example Problem

• Example 1: Solve the equation ( y' = -2y ) with initial condition ( y(0) = 1 ).

Step-by-Step Solution

1. Calculate initial slope: ( y' = -2y )
   • At ( y(0) = 1 ), slope = ( -2*1 = -2 )
2. First approximation step (h = 0.2):
   • ( y_1 = y_0 + h * y' = 1 + 0.2 * (-2) = 0.6 )
3. Second approximation step (h = 0.2):
   • With ( y_1 = 0.6 ), slope = ( -2*0.6 = -1.2 )
   • ( y_2 = y_1 + h * y' = 0.6 + 0.2 * (-1.2) = 0.36 )
4. Continue the method until required value/equation is achieved.

General Steps in Euler's Method

• Repeat the computation for each step by updating the current value and re-calculating the slope.
• Useful to validate intermediate and endpoint values to ensure accuracy.

Advantage and Limitation

• Advantage: Simple and easy to implement.
• Limitation: Accuracy depends on step size; smaller step size increases accuracy at the cost of more computations.
• Errors can accumulate, leading to less accurate results over large intervals.

Conclusion

• Euler's Method is a foundational technique in numerical analysis for approximating solutions to ODEs.
• Practical for initial value problems where analytical solutions are not feasible.

Practice Problems

1. Solve ( y' = x^2 - y ) with ( y(0) = 0 ) using step size ( h = 0.1 ) up to ( x = 1 ).
2. Solve ( y' = 3x + 2y ) with ( y(1) = 2 ) using step size ( h = 0.05 ) up to ( x = 2 ).