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Prof. Anderson 1. The Hermite equation y" - 2y +...
Consider the following differential equation. (1 + 5x2) y′′ − 8xy′ − 6y = 0 (a)...
Consider the following differential equation.
(1 + 5x2) y′′ − 8xy′
− 6y = 0
(a)
If you were to look for a power series solution about
x0 = 0, i.e., of the form
∞
Σ
n=0
cn xn
then the recurrence formula for the coefficients would be given by
ck+2 =
g(k) ck , k
≥ 2. Enter the function g(k) into the answer
box below.
(b)
Find the solution to the above differential equation with
initial conditions y(0) ...
Consider the following differential equation. (1 + 5x2)y" – 8xY' – 6y = 0 (a) If...
Consider the following differential equation. (1 + 5x2)y" – 8xY' – 6y = 0 (a) If you were to look for a power series solution about xo = 0, i.e., of the form Σ τη x2 n=0 then the recurrence formula for the coefficients would be given by C +2 g(k) Cx. k > 2. Enter the function g(k) into the answer box below. (b) Find the solution to the above differential equation with initial conditions (0) = 0 and...
In this exercise we consider finding the first five coefficients in the series solution of the first order linear initial value problem (+3)y' 2y 0 subject to the initial condition y(0) 1. Si...
In this exercise we consider finding the first five coefficients in the series solution of the first order linear initial value problem (+3)y' 2y 0 subject to the initial condition y(0) 1. Since the equation has an ordinary point at z 0 it has a power series solution in the form We learned how to easily solve problems like this separation of variables but here we want to consider the power series method (1) Insert the formal power series into...
Consider the following differential equation. (1 + 6x2)y" – 4xy' – 24y = 0 (a) If...
Consider the following differential equation. (1 + 6x2)y" – 4xy' – 24y = 0 (a) If you were to look for a power series solution about xo = 0, i.e., of the form onth n=0 then the recurrence formula for the coefficients would be given by ck+2 = g(k) Ck, k > 2. Enter the function g(k) into the answer box below. (b) Find the solution to the above differential equation with initial conditions y(0) = 0 and y'(0) =...
Solve the differential equation below with initial conditions. . Find the recurrence relation and compute the...
Solve the differential equation below with initial conditions. . Find the recurrence relation and compute the first 6 coefficients (a -a,) (1 3x)y y' 2xy 0 y(0) 1, y'(0)-0
cnrn Consider the following differential equation. (1 + 3x?) y" – 2xy' – 12y = 0...
cnrn Consider the following differential equation. (1 + 3x?) y" – 2xy' – 12y = 0 (a) If you were to look for a power series solution about xo = 0, i.e., of the form Σ n=0 00 then the recurrence formula for the coefficients would be given by Ck+2 g(k) Ck, k > 2. Enter the function g(k) into the answer box below. (b) Find the solution to the above differential equation with initial conditions y(0) = 0 and...
10.5.3 Consider the defining differential equation for the Hermite polynomials do and solve it by the series solution method for functions Hn(x such that Hx)exp(-x2/2) can be normalized In your solut...
10.5.3 Consider the defining differential equation for the Hermite polynomials do and solve it by the series solution method for functions Hn(x such that Hx)exp(-x2/2) can be normalized In your solution (i) find a recurrence relation between the coefficients of the power series solutions [Note: this (ii) show that Hn(x)exp(x/2) wll not be normalizable unless the power series terminates (ii) choosing co 0 or 1 and c0 or 1, find the first 5 power series solutions of the equation. relation...
5. Solve the linear, constant coefficient ODE y" – 3y' + 2y = 0; y(0) =...
5. Solve the linear, constant coefficient ODE y" – 3y' + 2y = 0; y(0) = 0, y'(0) = 1. 6. Solve the IVP with Cauchy-Euler ODE x2y" - 4xy' + 6y = 0; y(1) = 2, y'(1) = 0. 7. Given that y = Ge3x + cze-5x is a solution of the homogeneous equation, use the Method of Undetermined Coefficients to find the general solution of the non-homogeneous ODE " + 2y' - 15y = 3x 8. A 2...
Consider the following initial value problem, (1 - 2)" + 3xy' - 8y = 0, 3(0)...
Consider the following initial value problem, (1 - 2)" + 3xy' - 8y = 0, 3(0) = 3, 7(0) = 0. Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimals. (a) This differential equation has singular points at Note: You must use a semicolon here to separate your answers (b) Since there is no singular point at x = 0, you can find a normal power series solution for y() about...
Consider the following initial value problem, (1 - z2)y"+zy' - 12y-0, (0)3, y' (0)-0. Note: For each part b...
Consider the following initial value problem, (1 - z2)y"+zy' - 12y-0, (0)3, y' (0)-0. Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimals (a) This differential equation has singular points at Note: You must use a semicolon here to separate your answers. (b) Since there is no singular point at z 0, you can find a normal power series solution for y(x about z0,i.e. m-0 As part of the solution process...
Consider the following differential equation.
(1 + 5x2) y′′ − 8xy′
− 6y = 0
(a)
If you were to look for a power series solution about
x0 = 0, i.e., of the form
∞
Σ
n=0
cn xn
then the recurrence formula for the coefficients would be given by
ck+2 =
g(k) ck , k
≥ 2. Enter the function g(k) into the answer
box below.
(b)
Find the solution to the above differential equation with
initial conditions y(0) ...
Consider the following differential equation. (1 + 5x2)y" – 8xY' – 6y = 0 (a) If you were to look for a power series solution about xo = 0, i.e., of the form Σ τη x2 n=0 then the recurrence formula for the coefficients would be given by C +2 g(k) Cx. k > 2. Enter the function g(k) into the answer box below. (b) Find the solution to the above differential equation with initial conditions (0) = 0 and...
In this exercise we consider finding the first five coefficients in the series solution of the first order linear initial value problem (+3)y' 2y 0 subject to the initial condition y(0) 1. Since the equation has an ordinary point at z 0 it has a power series solution in the form We learned how to easily solve problems like this separation of variables but here we want to consider the power series method (1) Insert the formal power series into...
Consider the following differential equation. (1 + 6x2)y" – 4xy' – 24y = 0 (a) If you were to look for a power series solution about xo = 0, i.e., of the form onth n=0 then the recurrence formula for the coefficients would be given by ck+2 = g(k) Ck, k > 2. Enter the function g(k) into the answer box below. (b) Find the solution to the above differential equation with initial conditions y(0) = 0 and y'(0) =...
Solve the differential equation below with initial conditions. . Find the recurrence relation and compute the first 6 coefficients (a -a,) (1 3x)y y' 2xy 0 y(0) 1, y'(0)-0
cnrn Consider the following differential equation. (1 + 3x?) y" – 2xy' – 12y = 0 (a) If you were to look for a power series solution about xo = 0, i.e., of the form Σ n=0 00 then the recurrence formula for the coefficients would be given by Ck+2 g(k) Ck, k > 2. Enter the function g(k) into the answer box below. (b) Find the solution to the above differential equation with initial conditions y(0) = 0 and...
10.5.3 Consider the defining differential equation for the Hermite polynomials do and solve it by the series solution method for functions Hn(x such that Hx)exp(-x2/2) can be normalized In your solution (i) find a recurrence relation between the coefficients of the power series solutions [Note: this (ii) show that Hn(x)exp(x/2) wll not be normalizable unless the power series terminates (ii) choosing co 0 or 1 and c0 or 1, find the first 5 power series solutions of the equation. relation...
5. Solve the linear, constant coefficient ODE y" – 3y' + 2y = 0; y(0) = 0, y'(0) = 1. 6. Solve the IVP with Cauchy-Euler ODE x2y" - 4xy' + 6y = 0; y(1) = 2, y'(1) = 0. 7. Given that y = Ge3x + cze-5x is a solution of the homogeneous equation, use the Method of Undetermined Coefficients to find the general solution of the non-homogeneous ODE " + 2y' - 15y = 3x 8. A 2...
Consider the following initial value problem, (1 - 2)" + 3xy' - 8y = 0, 3(0) = 3, 7(0) = 0. Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimals. (a) This differential equation has singular points at Note: You must use a semicolon here to separate your answers (b) Since there is no singular point at x = 0, you can find a normal power series solution for y() about...
Consider the following initial value problem, (1 - z2)y"+zy' - 12y-0, (0)3, y' (0)-0. Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimals (a) This differential equation has singular points at Note: You must use a semicolon here to separate your answers. (b) Since there is no singular point at z 0, you can find a normal power series solution for y(x about z0,i.e. m-0 As part of the solution process...
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