Q. 58 Exercises 53-58, use Euler鈥檚 m... [FREE SOLUTION]

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Chapter 6: Q. 58 (page 571)

Exercises $53 - 58$ , use Euler鈥檚 method with the given $鈭� x$ to approximate four additional points on the graph of the solution $y (x)$ . Use these points to sketch a piecewise-linear approximation of the solution.
role="math" localid="1649302672566" $\frac{d y}{d x} = x^{2} - y, y (1) = 0; 鈭� x = 0.25$ .

Short Answer

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A graph for the piecewise-linear approximation of the solution for $\frac{d y}{d x} = x^{2} - y$ is,

Step by step solution

Step 1. Given information

$\frac{d y}{d x} = x^{2} - y, y (1) = 0; 鈭� x = 0.25$ .

Step 2. The aim is to approximate the solution of the differential equation dydx=x2-y that passes through the point x0,y0=1,0.

Use Euler's method to find four more points in the sequence by use of the iterative formula.

$(x_{k + 1}, y_{k + 1}) = (x_{k} + 鈭� x, y_{k} + 鈭� y_{k})$

Here $g (x, y) = x^{2} - y$ and $k = 0, 1, 2, 3$ . So, first find the four additional points by using above iterative formula.

$(x_{0}, y_{0}) = (1, 0) (x_{1}, y_{1}) = (x_{0} + 鈭� x, y_{0} + g (x_{0}, y_{0}) 鈭� x) = (1.25, 0 + 1 (0.25)) = (1.25, 0.25) (x_{2}, y_{2}) = (x_{1} + 鈭� x, y_{1} + g (x_{1}, y_{1}) 鈭� x) = (1.5, 0.25 + 1.3125 (0.25)) = (1.5, 0.5781) (x_{3}, y_{3}) = (x_{2} + 鈭� x, y_{2} + g (x_{2}, y_{2}) 鈭� x) = (1.75, 0.5781 + 1.6719 (0.25)) = (1.75, 0.996) (x_{4}, y_{4}) = (x_{3} + 鈭� x, y_{3} + g (x_{3}, y_{3}) 鈭� x) = (2, 0.996 + 2.0665 (0.25)) = (2, 1.5126)$

Step 3. Now plot the obtained five points and join the points with line segments as given below.

The approximate solution of the initial-value problem is shown by the graph by joining the line segments.

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Short Answer

Step by step solution

Step 1. Given information

Step 2. The aim is to approximate the solution of the differential equation dydx=x2-y that passes through the point x0,y0=1,0.

Step 3. Now plot the obtained five points and join the points with line segments as given below.

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91影视

Exercises 53-58, use Euler鈥檚 method with the given 鈭�xto approximate four additional points on the graph of the solution yx. Use these points to sketch a piecewise-linear approximation of the solution.role="math" localid="1649302672566" dydx=x2-y,y1=0;鈭�x=0.25.

Short Answer

Step by step solution

Step 1. Given information

Step 2. The aim is to approximate the solution of the differential equation dydx=x2-y that passes through the point x0,y0=1,0.

Step 3. Now plot the obtained five points and join the points with line segments as given below.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Calculus

Logic and Functions

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.