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The differential equation dy/dx= 0.25 y² is to be solved using the backward (implicit) Euler's method with the boundary condition y = 1 at x = 0 and with a step size of 1. What would be the value of y at x = 1?
(a) 1.33
(b) 1.67
(c)2.33
(d) 2.00?
(a) 1.33
(b) 1.67
(c)2.33
(d) 2.00?
Most Upvoted Answer
The differential equation dy/dx= 0.25 y² is to be solved using the bac...
Understanding the Problem
The differential equation given is dy/dx = 0.25 y² with the boundary condition y(0) = 1. We will use the backward (implicit) Euler's method to approximate the value of y at x = 1 with a step size of h = 1.
Backward Euler's Method
The backward Euler's method is defined as:
y_{n+1} = y_n + h * f(y_{n+1}, x_{n+1})
Here, f(y, x) = 0.25 y². For our case:
- y_0 = 1 (initial condition)
- x_0 = 0
- h = 1 (step size)
We need to compute y_1 at x = 1.
Setting Up the Equation
Using the backward Euler's formula at x = 1, we have:
y_1 = y_0 + h * f(y_1, 1)
Substituting the known values:
y_1 = 1 + 1 * 0.25 y_1²
This simplifies to:
y_1 = 1 + 0.25 y_1²
Rearranging the Equation
Rearranging gives us:
0.25 y_1² - y_1 + 1 = 0
Solving the Quadratic Equation
To solve for y_1, we can apply the quadratic formula:
y_1 = [1 ± sqrt(1² - 4 * 0.25 * 1)] / (2 * 0.25)
This leads us to:
y_1 = [1 ± sqrt(1 - 1)] / 0.5
y_1 = [1 ± 0] / 0.5
y_1 = 1 / 0.5 = 2
Conclusion
Thus, the value of y at x = 1 is 2. The correct answer is (d) 2.00.
The differential equation given is dy/dx = 0.25 y² with the boundary condition y(0) = 1. We will use the backward (implicit) Euler's method to approximate the value of y at x = 1 with a step size of h = 1.
Backward Euler's Method
The backward Euler's method is defined as:
y_{n+1} = y_n + h * f(y_{n+1}, x_{n+1})
Here, f(y, x) = 0.25 y². For our case:
- y_0 = 1 (initial condition)
- x_0 = 0
- h = 1 (step size)
We need to compute y_1 at x = 1.
Setting Up the Equation
Using the backward Euler's formula at x = 1, we have:
y_1 = y_0 + h * f(y_1, 1)
Substituting the known values:
y_1 = 1 + 1 * 0.25 y_1²
This simplifies to:
y_1 = 1 + 0.25 y_1²
Rearranging the Equation
Rearranging gives us:
0.25 y_1² - y_1 + 1 = 0
Solving the Quadratic Equation
To solve for y_1, we can apply the quadratic formula:
y_1 = [1 ± sqrt(1² - 4 * 0.25 * 1)] / (2 * 0.25)
This leads us to:
y_1 = [1 ± sqrt(1 - 1)] / 0.5
y_1 = [1 ± 0] / 0.5
y_1 = 1 / 0.5 = 2
Conclusion
Thus, the value of y at x = 1 is 2. The correct answer is (d) 2.00.
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The differential equation dy/dx= 0.25 y² is to be solved using the backward (implicit) Euler's method with the boundary condition y = 1 at x = 0 and with a step size of 1. What would be the value of y at x = 1?(a) 1.33(b) 1.67(c)2.33(d) 2.00?
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The differential equation dy/dx= 0.25 y² is to be solved using the backward (implicit) Euler's method with the boundary condition y = 1 at x = 0 and with a step size of 1. What would be the value of y at x = 1?(a) 1.33(b) 1.67(c)2.33(d) 2.00? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about The differential equation dy/dx= 0.25 y² is to be solved using the backward (implicit) Euler's method with the boundary condition y = 1 at x = 0 and with a step size of 1. What would be the value of y at x = 1?(a) 1.33(b) 1.67(c)2.33(d) 2.00? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The differential equation dy/dx= 0.25 y² is to be solved using the backward (implicit) Euler's method with the boundary condition y = 1 at x = 0 and with a step size of 1. What would be the value of y at x = 1?(a) 1.33(b) 1.67(c)2.33(d) 2.00?.
The differential equation dy/dx= 0.25 y² is to be solved using the backward (implicit) Euler's method with the boundary condition y = 1 at x = 0 and with a step size of 1. What would be the value of y at x = 1?(a) 1.33(b) 1.67(c)2.33(d) 2.00? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about The differential equation dy/dx= 0.25 y² is to be solved using the backward (implicit) Euler's method with the boundary condition y = 1 at x = 0 and with a step size of 1. What would be the value of y at x = 1?(a) 1.33(b) 1.67(c)2.33(d) 2.00? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The differential equation dy/dx= 0.25 y² is to be solved using the backward (implicit) Euler's method with the boundary condition y = 1 at x = 0 and with a step size of 1. What would be the value of y at x = 1?(a) 1.33(b) 1.67(c)2.33(d) 2.00?.
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