Homework Answers
6. (6+2=8pts) Consider the ODE (2 - xy + y = 0. (a) Assuming a power...
6. (6+2=8pts) Consider the ODE (2 - xy + y = 0. (a) Assuming a power series solution of the form y = -Ź anz", find a recurrence relation that the coefficients must satisfy. NO (b) Using the recurrence relation in part (a), express the coefficients az and az in terms of ao and ai
Problem #2: Consider the following statements. [6 marks) (1) The particular solution of the ODE)" -...
Problem #2: Consider the following statements. [6 marks) (1) The particular solution of the ODE)" - 6y' + 9y = 5e3x is given by yp = Cre3x where C is an undetermined constant. (ii) The procedure of finding series solutions to a homogeneous linear second-order ODEs could be accurately described as the "method of undetermined series coefficients". (iii) Most of the material in Lecture Notes from Week 3 to Week 5, inclusive, can be extended or generalized to higher-order ODES...
Consider the ODE:3xy"+y' - 2xy = 0. Find the general solution in power series form about...
Consider the ODE:3xy"+y' - 2xy = 0. Find the general solution in power series form about the regular singular point x = 0, following parts (a) – (c), below. (a) Obtain the recurrence relation. (b) Find the exponents of the singularity. (e) Obtain only one of the two linearly independent solutions, call it y(x), that corresponds to the smaller exponent of the singularity; but, only explicitly include the first four non-zero terms of the power series solution. Write down the...
(1 point) Find the indicated coefficients of the power series solution about x-0 of the differential equation (x2-x+1y"-y-3y = 0, y(0) = 0, y(o) =-8 x2+ 4 (1 point) Find the indicated coeffi...
(1 point) Find the indicated coefficients of the power series solution about x-0 of the differential equation (x2-x+1y"-y-3y = 0, y(0) = 0, y(o) =-8 x2+ 4
(1 point) Find the indicated coefficients of the power series solution about x-0 of the differential equation (x2-x+1y"-y-3y = 0, y(0) = 0, y(o) =-8 x2+ 4
Find the indicated coefficients of the power series solution about x=0 of the differential equation. (x^2+1)y''-...
Find the indicated coefficients of the power series solution
about x=0 of the differential equation.
(x^2+1)y''-xy'+y=0, y(0)=3, y'(0)=-6
(1 point) Find the indicated coefficients of the power series solution about 0 of the differential equation (x2 1)y ry y 0, (0) 3, y' (0) -6 r2 24+ r(9)
(1 point) Find the indicated coefficients of the power series solution about 0 of the differential equation (x2 1)y ry y 0, (0) 3, y' (0) -6 r2 24+ r(9)
3. Consider the following ODE: (1 + 2%)/" - xy + y = 0 (a) Find...
3. Consider the following ODE: (1 + 2%)/" - xy + y = 0 (a) Find the first 3 nonzero terms of the power series expansion (around x = 0) for the general solution. (b) Use the ratio test to determine the radius of convergence of the series. What can you say about the radius of convergence without solving the ODE? (c) Determine the solution that satisfies the initial conditions y(0) = 1 and (0) = 0.
Please answer (i) (ii) and (iii) 5. For each of the following linear homogeneous ODE, do...
Please answer (i) (ii) and
(iii)
5. For each of the following linear homogeneous ODE, do the fol- lowing. (a) Identify p(x) and (2) and, from them, determine the least possible guaranteed interval of convergence about the specified center Jo. (b) Write the general solution in the form of a power series, obtaining the first three non-vanishing powers of (x - Xo) in each of yı (2) and y2(2). 2- +y=0, = 1 y" + cos(x)y=0, x = 0 15"...
Question 3 Consider the ordinary differential equation (ODE) 2xy" + (1 + x)y' + 3y =...
Question 3 Consider the ordinary differential equation (ODE) 2xy" + (1 + x)y' + 3y = 0, in the neighbourhood of the origin. a) Show that x = 0 is a regular singular point of the ODE. (10) b) By seeking an appropriate solution to the ODE, show that G=- (10) i) the roots to the indicial equation of the ODE are 0 and 1/2. [10] ii) the recurrence formula used to determine the power series coefficients, ens when one...
Question 1 4 pts To find a power series solution about x = 0 to y...
Question 1 4 pts To find a power series solution about x = 0 to y + 2xy = 0, which are procedures needed? Apply the Theorem 3 that all coefficients must be O to determine the coefficients an Show x = 0 is an ordinary point. Shift the indices so that the general term in each is a constant times ck and combined these power series as only one series. All of them Write the solution as a power...
(4) Non-Constant Coefficient ODE: Consider the ODEs below for a dependent variable y(x). Solve them in...
(4) Non-Constant Coefficient ODE: Consider the ODEs below for a dependent variable y(x). Solve them in general (no application of ICs or BCs) using power series methods, letting y(x) = n=oxn+r and including the power r is necessary. See the class notes of 9/23/2019. i) y" + way = 0, ii) y" – xy = 0.
6. (6+2=8pts) Consider the ODE (2 - xy + y = 0. (a) Assuming a power series solution of the form y = -Ź anz", find a recurrence relation that the coefficients must satisfy. NO (b) Using the recurrence relation in part (a), express the coefficients az and az in terms of ao and ai
Problem #2: Consider the following statements. [6 marks) (1) The particular solution of the ODE)" - 6y' + 9y = 5e3x is given by yp = Cre3x where C is an undetermined constant. (ii) The procedure of finding series solutions to a homogeneous linear second-order ODEs could be accurately described as the "method of undetermined series coefficients". (iii) Most of the material in Lecture Notes from Week 3 to Week 5, inclusive, can be extended or generalized to higher-order ODES...
Consider the ODE:3xy"+y' - 2xy = 0. Find the general solution in power series form about the regular singular point x = 0, following parts (a) – (c), below. (a) Obtain the recurrence relation. (b) Find the exponents of the singularity. (e) Obtain only one of the two linearly independent solutions, call it y(x), that corresponds to the smaller exponent of the singularity; but, only explicitly include the first four non-zero terms of the power series solution. Write down the...
(1 point) Find the indicated coefficients of the power series solution about x-0 of the differential equation (x2-x+1y"-y-3y = 0, y(0) = 0, y(o) =-8 x2+ 4
(1 point) Find the indicated coefficients of the power series solution about x-0 of the differential equation (x2-x+1y"-y-3y = 0, y(0) = 0, y(o) =-8 x2+ 4
Find the indicated coefficients of the power series solution
about x=0 of the differential equation.
(x^2+1)y''-xy'+y=0, y(0)=3, y'(0)=-6
(1 point) Find the indicated coefficients of the power series solution about 0 of the differential equation (x2 1)y ry y 0, (0) 3, y' (0) -6 r2 24+ r(9)
(1 point) Find the indicated coefficients of the power series solution about 0 of the differential equation (x2 1)y ry y 0, (0) 3, y' (0) -6 r2 24+ r(9)
3. Consider the following ODE: (1 + 2%)/" - xy + y = 0 (a) Find the first 3 nonzero terms of the power series expansion (around x = 0) for the general solution. (b) Use the ratio test to determine the radius of convergence of the series. What can you say about the radius of convergence without solving the ODE? (c) Determine the solution that satisfies the initial conditions y(0) = 1 and (0) = 0.
Please answer (i) (ii) and
(iii)
5. For each of the following linear homogeneous ODE, do the fol- lowing. (a) Identify p(x) and (2) and, from them, determine the least possible guaranteed interval of convergence about the specified center Jo. (b) Write the general solution in the form of a power series, obtaining the first three non-vanishing powers of (x - Xo) in each of yı (2) and y2(2). 2- +y=0, = 1 y" + cos(x)y=0, x = 0 15"...
Question 3 Consider the ordinary differential equation (ODE) 2xy" + (1 + x)y' + 3y = 0, in the neighbourhood of the origin. a) Show that x = 0 is a regular singular point of the ODE. (10) b) By seeking an appropriate solution to the ODE, show that G=- (10) i) the roots to the indicial equation of the ODE are 0 and 1/2. [10] ii) the recurrence formula used to determine the power series coefficients, ens when one...
Question 1 4 pts To find a power series solution about x = 0 to y + 2xy = 0, which are procedures needed? Apply the Theorem 3 that all coefficients must be O to determine the coefficients an Show x = 0 is an ordinary point. Shift the indices so that the general term in each is a constant times ck and combined these power series as only one series. All of them Write the solution as a power...
(4) Non-Constant Coefficient ODE: Consider the ODEs below for a dependent variable y(x). Solve them in general (no application of ICs or BCs) using power series methods, letting y(x) = n=oxn+r and including the power r is necessary. See the class notes of 9/23/2019. i) y" + way = 0, ii) y" – xy = 0.
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