Question

Consider the differential equation 4x2y′′ − 8x2y′ + (4x2 + 1)y = 0 (a) Verify that...

Consider the differential equation

4x2y′′ − 8x2y′ + (4x2 + 1)y = 0

(a) Verify that x0 = 0 is a regular singular point of the differential equation and then find one solution as a Frobenius series centered at x0 = 0. The indicial equation has a single root with multiplicity two. Therefore the differential equation has only one Frobenius series solution. Write your solution in terms of familiar elementary functions.

(b) Use Reduction of Order to find a second linearly independent solution and then form the general solution of the differential equation.

Note: In part (a), you will find one solution of the differential equation is y = √xex. You must show all of the work of finding the series in part (a), but if you cannot, proceed to reduction of order with the given function.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Given that x =0 is a regular singular point of the given differential equation, show that...
Given that x =0 is a regular singular point of the given differential equation, show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain to linearly independent series solutions about x = 0. Form the general solution on (0, ∞) 3xy”+(2 – x)y’ – y = 0
7. Given that x =0 is a regular singular point of the given differential equation, show...
7. Given that x =0 is a regular singular point of the given differential equation, show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on (0, ∞) 2xy”- y’ + y = 0
Consider the differential equation x^2 y' '+ x^2 y' + (x-2)y = 0 a) Show that...
Consider the differential equation x^2 y' '+ x^2 y' + (x-2)y = 0 a) Show that x = 0 is a regular singular point for the equation. b) For a series solution of the form y = ∑∞ n=0 an x^(n+r)   a0 ̸= 0 of the differential equation about x = 0, find a recurrence relation that defines the coefficients an’s corresponding to the larger root of the indicial equation. Do not solve the recurrence relation.
Consider the differential equation x2y''+xy'-y=0, x>0. a. Verify that y(x)=x is a solution. b. Find a...
Consider the differential equation x2y''+xy'-y=0, x>0. a. Verify that y(x)=x is a solution. b. Find a second linearly independent solution using the method of reduction of order. [Please use y2(x) = v(x)y1(x)]
consider the differential equation xy'' - xy' + y = 0. The indicial equation is r(r...
consider the differential equation xy'' - xy' + y = 0. The indicial equation is r(r - 1) = 0. The recurrence relation is c_k+1(k + r + 1) + (k + r) - c_k(K + r - 1) = 0. A series solution to the indicial root r = 0 is
Consider the following differential equation 32x 2y '' + 3 (1 − e 2x )y =...
Consider the following differential equation 32x 2y '' + 3 (1 − e 2x )y = 0 (b) Determine the indicial equation and find its roots. (c) Without solving the problem, formally write the two linearly independent solutions near x = 0. (d) What can you say about the radius of convergence of the power series in (c)? (e) Find the first three non-zero terms of the two linearly independent solutions.
Series Solutions of Ordinary Differential Equations For the following problems solve the given differential equation by...
Series Solutions of Ordinary Differential Equations For the following problems solve the given differential equation by means of a power series about the given point x0. Find the recurrence relation; also find the first four terms in each of two linearly independed sollutions (unless the series terminates sooner). If possible, find the general term in each solution. y"+k2x2y=0, x0=0, k-constant
Differential Equation: Determine two linearly independent power series solutions centered at x=0. y” - x^2 y’...
Differential Equation: Determine two linearly independent power series solutions centered at x=0. y” - x^2 y’ - 2xy = 0
Series Solution Method. Solve the given differential equation by means of a power series about the...
Series Solution Method. Solve the given differential equation by means of a power series about the given point x0. Find the recurrence relation; also find the first four terms in each of two linearly independent solutions (unless the series terminates sooner). If possible, find the general term in each solution. (1 − x)y′′ + y = 0, x0 = 0
The point x = 0 is a regular singular point of the differential equation. x^2y'' +...
The point x = 0 is a regular singular point of the differential equation. x^2y'' + (9 /5 x + x^2) y' − 1/ 5 y = 0. Use the general form of the indicial equation (14) in Section 6.3 r(r − 1) + a0 r + b0 = 0 (14) to find the indicial roots of the singularity. (List the indicial roots below as a comma-separated list.) r =
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT