Question

3. (a) Given the initial value problem y''(x) = 2(1 + 2x^2)y(x), x_0 = 0, y(x_0) = 1, y'(x_0) = 0 rewrite the second-order ordinary differential equation as a set of two coupled first-order ODEs. i.e. if we set egin{pmatrix} y_1(x) \ y_2(x) end{pmatrix} = egin{pmatrix} y(x) \ y'(x) end{pmatrix} and then construct the set of coupled first-order ODEs y'_i(x) = f_i(x, y_1, y_2), i = 1, 2 what are the forms of f_1 and f_2? [2 marks] (b) Use Euler's method to calculate y(x_1) with x_1 = x_0 + Delta x and Delta x = 1. [2 marks] (c) In practice, why is Euler's method generally not used to solve initial value problems? [1 mark] (d) The simplest Runge Kutta algorithm uses information at a half step in making a full step. y_i(x + Delta x) = y_i(x) + Delta x f_i(x + Delta x/2, y_1(x + Delta x/2), y_2(x + Delta x/2)), i = 1, 2 where y_i(x + Delta x/2), i = 1, 2 is given by an Euler step. Recalculate y(x_1) using the Runge-Kutta method. [3 marks]

          3. (a) Given the initial value problem
y''(x) = 2(1 + 2x^2)y(x), x_0 = 0, y(x_0) = 1, y'(x_0) = 0
rewrite the second-order ordinary differential equation as a set of two coupled first-order ODEs. i.e. if we set
egin{pmatrix} y_1(x) \ y_2(x) end{pmatrix} = egin{pmatrix} y(x) \ y'(x) end{pmatrix}
and then construct the set of coupled first-order ODEs
y'_i(x) = f_i(x, y_1, y_2), i = 1, 2
what are the forms of f_1 and f_2? [2 marks]
(b) Use Euler's method to calculate y(x_1) with x_1 = x_0 + Delta x and Delta x = 1. [2 marks]
(c) In practice, why is Euler's method generally not used to solve initial value problems? [1 mark]
(d) The simplest Runge Kutta algorithm uses information at a half step in making a full step.
y_i(x + Delta x) = y_i(x) + Delta x f_i(x + Delta x/2, y_1(x + Delta x/2), y_2(x + Delta x/2)), i = 1, 2
where
y_i(x + Delta x/2), i = 1, 2
is given by an Euler step.
Recalculate y(x_1) using the Runge-Kutta method. [3 marks]

3. (a) Given the initial value problem
y”(x) = 2(1 + 2x^2)y(x), x0 = 0, y(x0) = 1, y'(x0) = 0
rewrite the second-order ordinary differential equation as a set of two coupled first-order ODEs. i.e. if we set
eginpmatrix y1(x) y2(x) endpmatrix = eginpmatrix y(x) y'(x) endpmatrix
and then construct the set of coupled first-order ODEs
y'i(x) = fi(x, y1, y2), i = 1, 2
what are the forms of f1 and f2? [2 marks]
(b) Use Euler's method to calculate y(x1) with x1 = x0 + Delta x and Delta x = 1. [2 marks]
(c) In practice, why is Euler's method generally not used to solve initial value problems? [1 mark]
(d) The simplest Runge Kutta algorithm uses information at a half step in making a full step.
yi(x + Delta x) = yi(x) + Delta x fi(x + Delta x/2, y1(x + Delta x/2), y2(x + Delta x/2)), i = 1, 2
where
yi(x + Delta x/2), i = 1, 2
is given by an Euler step.
Recalculate y(x1) using the Runge-Kutta method. [3 marks]

Added by Wendy B.

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