Short Answer

In Section 3.6, we discussed the improved Euler’s method for approximating the solution to a first-order equation. Extend this method to normal systems and give the recursive formulas for solving the initial value problem.

The result is:

$x_{i, n + 1} = x_{i, n} + \frac{h}{2} [f_{i} (t_{n}, x_{1, n}, x_{2, n} . . . . . . . ., x_{m, n}) + f_{i} (t_{n} + h, x_{1, n} + {hf}_{1} (t_{n}, x_{1, n}, x_{2, n} . . . . . . . ., x_{m, n}) . . . . . . . ., x_{m, n} + {hf}_{m} (t_{n}, x_{1, n}, x_{2, n} . . . . . . . ., x_{m, n})]$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Use Euler’s method

Here given Euler’s method of the differential equation:

So, $y_{n + 1} = y_{n} + \frac{h}{2} [f (x_{n}, y_{n}) + f (x_{n} + h, y_{n} + hf (x_{n}, y_{n}))]$

n=0,1,2,…..and $x_{n + 1} = x_{n} + h$ .

Now,

$x'_{1} (t) = f_{1} (t, x_{1}, x_{2} . . . . . . . ., x_{m}) x'_{2} (t) = f_{2} (t, x_{1}, x_{2} . . . . . . . ., x_{m}) . . x'_{m} (t) = f_{m} (t, x_{1}, x_{2} . . . . . . . ., x_{m})$

Step 2: Solve for every i

For every I from 1 to m, then;

$t_{n + 1} = t_{n} + h x_{i, n + 1} = x_{i, n} + \frac{h}{2} [f_{i} (t_{n}, x_{1, n}, x_{2, n} . . . . . . . ., x_{m, n}) + f_{i} (t_{n} + h, x_{1, n} + {hf}_{1} (t_{n}, x_{1, n}, x_{2, n} . . . . . . . ., x_{m, n}) . . . . . . . ., x_{m, n} + {hf}_{m} (t_{n}, x_{1, n}, x_{2, n} . . . . . . . ., x_{m, n})]$

This is the required result.

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

In Section 3.6, we discussed the improved Euler’s method for approximating the solution to a first-order equation. Extend this method to normal systems and give the recursive formulas for solving the initial value problem.

Step by Step Solution

Step 1: Use Euler’s method

Step 2: Solve for every i

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In Section 3.6, we discussed the improved Euler’s method for approximating the solution to a first-order equation. Extend this method to normal systems and give the recursive formulas for solving the initial value problem.

Step by Step Solution

Step 1: Use Euler’s method

Step 2: Solve for every i

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