To apply Simpson's 3/8 rule, the appropriate value of n = ___. Find y(1.2) using Euler's method for the differential equation dy/dx = xy; y(1.1) = 2.1 (Use h = 0.1). Solve an initi... To apply Simpson's 3/8 rule, the appropriate value of n = ___. Find y(1.2) using Euler's method for the differential equation dy/dx = xy; y(1.1) = 2.1 (Use h = 0.1). Solve an initial value problem y' = x + y; y(1) = 2 for y = 1.1 using Kutta method of order 4. (Take h = 0.1)
Understand the Problem
The question involves two parts: first, it asks to determine the appropriate value of 'n' for Simpson's 3/8 rule and secondly, to find the value of y(1.2) using Euler's method for a given differential equation. It also requests a solution for an initial value problem using the Runge-Kutta method of order 4.
Answer
The appropriate value of \( n \) is \( 6 \), \( y(1.2) = 2.331 \), and \( y(1.1) \approx 2.3216 \).
Answer for screen readers
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The appropriate value of ( n ) for Simpson's 3/8 rule is ( 6 ).
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The value of ( y(1.2) ) using Euler's method is ( 2.331 ).
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The value of ( y(1.1) ) using Runge-Kutta method of order 4 is approximately ( 2.3216 ).
Steps to Solve
- Determine the value of n for Simpson's 3/8 Rule
For Simpson's 3/8 rule, the value of ( n ) must be a multiple of 3, since the rule is derived for equally spaced segments of 3 points. The possible values given are 7, 8, 11, and 6. The closest suitable value of ( n ) from the options provided is 6 (as it is a multiple of 3).
- Apply Euler’s Method to find ( y(1.2) )
Using the differential equation ( \frac{dy}{dx} = xy ) with initial condition ( y(1.1) = 2.1 ) and step size ( h = 0.1 ):
The formula for Euler's method is: $$ y_{n+1} = y_n + h f(x_n, y_n) $$ Where ( f(x, y) = xy ).
- Calculate ( y(1.2) ):
- For ( x_0 = 1.1 ), ( y_0 = 2.1 )
- ( f(1.1, 2.1) = 1.1 \times 2.1 = 2.31 )
- Update using Euler: $$ y(1.2) = 2.1 + 0.1 \times 2.31 = 2.1 + 0.231 = 2.331 $$
- Use the Runge-Kutta Method of Order 4
We need to solve the problem ( y' = x + y ) with initial condition ( y(1) = 2 ) and ( h = 0.1 ).
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For ( x_0 = 1 ), ( y_0 = 2 ):
Calculate the next ( y ):
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Calculate ( k_1 ): $$ k_1 = h f(x_0, y_0) = 0.1(1 + 2) = 0.3 $$
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Calculate ( k_2 ): $$ k_2 = h f\left(x_0 + \frac{h}{2}, y_0 + \frac{k_1}{2}\right) = 0.1\left(1.05 + 2.15\right) = 0.315 $$
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Calculate ( k_3 ): $$ k_3 = h f\left(x_0 + \frac{h}{2}, y_0 + \frac{k_2}{2}\right) = 0.1\left(1.05 + 2.1575\right) = 0.31575 $$
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Calculate ( k_4 ): $$ k_4 = h f(x_0 + h, y_0 + k_3) = 0.1(1.1 + 2.31575) = 0.341575 $$
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Combine to find ( y(1.1) ): $$ y(1.1) = y_0 + \frac{k_1 + 2k_2 + 2k_3 + k_4}{6} = 2 + \frac{0.3 + 2(0.315) + 2(0.31575) + 0.341575}{6} = 2 + 0.321618 = 2.321618 $$
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The appropriate value of ( n ) for Simpson's 3/8 rule is ( 6 ).
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The value of ( y(1.2) ) using Euler's method is ( 2.331 ).
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The value of ( y(1.1) ) using Runge-Kutta method of order 4 is approximately ( 2.3216 ).
More Information
- Simpson's 3/8 rule applies to equally spaced intervals, and ( n ) must be a multiple of 3.
- Euler’s method is a first-order numerical method for solving ordinary differential equations.
- The Runge-Kutta method of order 4 is a popular technique for obtaining numerical solutions due to its accuracy.
Tips
- Misunderstanding the requirement for ( n ) in Simpson's 3/8 rule could lead to selecting the wrong option.
- Failing to account for the step size ( h ) in Euler's method could lead to incorrect ( y ) values.
- Using incorrect formulas or coefficients while applying the Runge-Kutta method can yield inaccurate results.
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