Chapter 3: Q 3.7-4E (page 139)

Determine the recursive formulas for the Taylor method of order 4 for the initial value problem $y' = x^{2} + y, y (0) = 0$ .

Short Answer

Expert verified

$y_{n + 1} = y_{n} + h ({x_{n}}^{2} + y_{n}) + \frac{h^{2}}{2} (2 x + x^{2} + y) + \frac{h^{3}}{6} (2 + 2 x + x^{2} + y) + \frac{h^{4}}{24} (2 + 2 x + x^{2} + y)$

Step by step solution

Find the value of f2(x,y)

Here $y' = x^{2} + y, y (0) = 0$

Apply the chain rule.

$f_{2} (x, y) = \frac{\partial f}{\partial x} (x, y) + \frac{\partial f}{\partial y} (x, y) f (x, y)$

Since ${f(x,y) =x}^{2} + y$

$\frac{\partial f}{\partial x} (x, y) = 2 x \frac{\partial f}{\partial y} (x, y) = 1$

So, the equation is $f_{2} (x, y) = 2 x + x^{2} + y$

Evaluate the values of f3(x,y) and f4 (x,y)

Apply the same procedure as step 1

$f_{3} (x, y) = 2 + 2 x + x^{2} + y f_{4} (x, y) = 2 + 2 x + x + y$

Apply the recursive formulas for order 4

The recursive formula is

$x_{n + 1} = x_{n} + h y_{n + 1} = y_{n} + hf (x_{n} + y_{n}) + \frac{h^{2}}{2!} f_{2} (x_{n} + y_{n}) + . . . . . \frac{h^{p}}{p!} f_{p} (x_{n} + y_{n})$

$x_{n + 1} = x_{n} + h y_{n + 1} = y_{n} + h ({x_{n}}^{2} + y_{n}) + \frac{h^{2}}{2} (2 x + x^{2} + y) + \frac{h^{3}}{6} (2 + 2 x + x^{2} + y) + \frac{h^{4}}{24} (2 + 2 x + x^{2} + y)$

Where starting points are $x_{o} = 0, y_{0} = 0$ .

Hence the solution is

role="math" localid="1664317215716" $y_{n + 1} = y_{n} + h ({x_{n}}^{2} + y_{n}) + \frac{h^{2}}{2} (2 x + x^{2} + y) + \frac{h^{3}}{6} (2 + 2 x + x^{2} + y) + \frac{h^{4}}{24} (2 + 2 x + x^{2} + y)$

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Determine the recursive formulas for the Taylor method of order 4 for the initial value problem $y' = x^{2} + y, y (0) = 0$ .

Short Answer

Step by step solution

Find the value of f2(x,y)

Evaluate the values of f3(x,y) and f4 (x,y)

Apply the recursive formulas for order 4

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Determine the recursive formulas for the Taylor method of order 4 for the initial value problem y'=x2+y,y(0)=0.

Short Answer

Step by step solution

Find the value of f2(x,y)

Evaluate the values of f3(x,y) and f4 (x,y)

Apply the recursive formulas for order 4

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Mechanics Maths

Geometry

Discrete Mathematics

Decision Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

Determine the recursive formulas for the Taylor method of order 4 for the initial value problem $y' = x^{2} + y, y (0) = 0$ .