Homework Answers
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.
Find the first five terms of the series solution to the IVP (y +(1-2) +2y=e", y(0)...
Find the first five terms of the series solution to the IVP (y +(1-2) +2y=e", y(0) = -5, (y0 =1, by making use of the general power series representation in (2). Hint: Recall the Taylor/power series for et about the point 0.
1) Find the first 5-term in the power series general solution to the DE: Y" –...
1) Find the first 5-term in the power series general solution to the DE: Y" – xy' +2y = 0
Q2. Solve xy" + y' + 9xy = 0 in terms of Bessel functions. Q3. Consider...
Q2. Solve xy" + y' + 9xy = 0 in terms of Bessel functions. Q3. Consider the power series S. (x) = 2 0 (1) Find the radius of convergence for S..(x). (ii) If S. (x) is the series expansion for f(x) = S..(x)? Explain. [6] Can we compute f(12) by replacing x by 12 in [4]
3. For the differential equation (2x2 - 1)/" + xy + 2y = 0, find the...
3. For the differential equation (2x2 - 1)/" + xy + 2y = 0, find the first three non-zero terms of each of two linearly independent power series solutions. 4. Find the general solution of the system of equations
Name: 3) Bessel's Functions. Consider the differential equation y xy+y- power series solution of ...
Name: 3) Bessel's Functions. Consider the differential equation y xy+y- power series solution of y +xy+y- Section: 003 402 404 406 a) Use the method of Frobenius (which we learned in 7.3) to find a recurrence relation for the b) Find a general form of the answer, using only factorials (not the Gamma function), c) Determine the radius of convergence of your power series answer d) This is called a Bessel function of order zero. What is the differential equation...
15. (1) Find a power series solution of the differential equation. (2) Determine the radius of convergence of the resulting series. (3) Identify the series solution in terms of familiar elementary fu...
15. (1) Find a power series solution of the differential equation. (2) Determine the radius of convergence of the resulting series. (3) Identify the series solution in terms of familiar elementary functions when possible. No credit for any other methods. (x2+1)' + 3xy' +2y=0
15. (1) Find a power series solution of the differential equation. (2) Determine the radius of convergence of the resulting series. (3) Identify the series solution in terms of familiar elementary functions when possible. No credit...
3.2 Problems Find general solutions in powers of x of the diferential equa- tions in Problems 1 t...
5 please
and 17 only
3.2 Problems Find general solutions in powers of x of the diferential equa- tions in Problems 1 through 15. State the recurrence relation and the guaranteed radius of convergence in each case. 1, (x2-1 )y', + 4xy' + 2y = 0 2. (x2 + 2)y', + 4xy' + 2y = 0 3. y+xy y 0 4. (x2 + 1)y', + 6xy' + 4y = 0 5. (x2 3)y' +2xy 0 Use power series to solve...
3. Find the general solution to the differential equation y"2y 0 as a power series about 0 involving two free const...
3. Find the general solution to the differential equation y"2y 0 as a power series about 0 involving two free constants; the formula for the nth coefficient need not be in 'simplified' form.
Xy', + y,-xy = 0, x,-0 Answer the following questions a) What are the points of singularity for e...
xy', + y,-xy = 0, x,-0 Answer the following questions a) What are the points of singularity for each specific problem? b) Does this ODE hold a general power series solution at the specific x0? Justify your answer, i) if your answer is yes, the proceed as follows: Compute the radius of analyticity and report the corresponding interval, and ldentify the recursion formula for the power series coefficient around x0, and write the corresponding solution with at least four non-zero...
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.
Find the first five terms of the series solution to the IVP (y +(1-2) +2y=e", y(0) = -5, (y0 =1, by making use of the general power series representation in (2). Hint: Recall the Taylor/power series for et about the point 0.
1) Find the first 5-term in the power series general solution to the DE: Y" – xy' +2y = 0
Q2. Solve xy" + y' + 9xy = 0 in terms of Bessel functions. Q3. Consider the power series S. (x) = 2 0 (1) Find the radius of convergence for S..(x). (ii) If S. (x) is the series expansion for f(x) = S..(x)? Explain. [6] Can we compute f(12) by replacing x by 12 in [4]
3. For the differential equation (2x2 - 1)/" + xy + 2y = 0, find the first three non-zero terms of each of two linearly independent power series solutions. 4. Find the general solution of the system of equations
Name: 3) Bessel's Functions. Consider the differential equation y xy+y- power series solution of y +xy+y- Section: 003 402 404 406 a) Use the method of Frobenius (which we learned in 7.3) to find a recurrence relation for the b) Find a general form of the answer, using only factorials (not the Gamma function), c) Determine the radius of convergence of your power series answer d) This is called a Bessel function of order zero. What is the differential equation...
15. (1) Find a power series solution of the differential equation. (2) Determine the radius of convergence of the resulting series. (3) Identify the series solution in terms of familiar elementary functions when possible. No credit for any other methods. (x2+1)' + 3xy' +2y=0
15. (1) Find a power series solution of the differential equation. (2) Determine the radius of convergence of the resulting series. (3) Identify the series solution in terms of familiar elementary functions when possible. No credit...
5 please
and 17 only
3.2 Problems Find general solutions in powers of x of the diferential equa- tions in Problems 1 through 15. State the recurrence relation and the guaranteed radius of convergence in each case. 1, (x2-1 )y', + 4xy' + 2y = 0 2. (x2 + 2)y', + 4xy' + 2y = 0 3. y+xy y 0 4. (x2 + 1)y', + 6xy' + 4y = 0 5. (x2 3)y' +2xy 0 Use power series to solve...
3. Find the general solution to the differential equation y"2y 0 as a power series about 0 involving two free constants; the formula for the nth coefficient need not be in 'simplified' form.
xy', + y,-xy = 0, x,-0 Answer the following questions a) What are the points of singularity for each specific problem? b) Does this ODE hold a general power series solution at the specific x0? Justify your answer, i) if your answer is yes, the proceed as follows: Compute the radius of analyticity and report the corresponding interval, and ldentify the recursion formula for the power series coefficient around x0, and write the corresponding solution with at least four non-zero...
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