- Please Answer All Thanks 1 (48.49 KiB) Viewed 31 times
PLEASE ANSWER ALL thanks.
-
answerhappygod
- Site Admin
- Posts: 899604
- Joined: Mon Aug 02, 2021 8:13 am
PLEASE ANSWER ALL thanks.
PLEASE ANSWER ALL thanks.
Consider the variable coefficient linear second order homogeneous ODE (r²+1)y" - 4ry + 6y=0. (1) 2 1. The point r=0 is an ordinary point of equation (1). Therefore, we can find a power series solution of the form. y = Σama™ m=0 Write down the first and second derivatives of the power series. 2. Substitute the power series (and its derivatives) into equation (1). Express your answer in the form DO m-2 Σbmx" +Σ²2™ = 0, m=0 where bm and em are to be written in terms of m and am 3. Shift the index on one of the series you found in 2 so that the exponents of r are equal to m in both series. 2 4. Find a recurrence relation for the coefficients am+2 in terms of am and m. 5. Use the recurrence relation to find expressions for the coefficients az, as, a4 and as. 3 6. Write down the general solution to (1) in the form y = aof (x) + a19(2). 7. Find the particular solution of (1) that satisfies the initial conditions y(0) = 3 and y'(0) = 2.
Consider the variable coefficient linear second order homogeneous ODE (r²+1)y" - 4ry + 6y=0. (1) 2 1. The point r=0 is an ordinary point of equation (1). Therefore, we can find a power series solution of the form. y = Σama™ m=0 Write down the first and second derivatives of the power series. 2. Substitute the power series (and its derivatives) into equation (1). Express your answer in the form DO m-2 Σbmx" +Σ²2™ = 0, m=0 where bm and em are to be written in terms of m and am 3. Shift the index on one of the series you found in 2 so that the exponents of r are equal to m in both series. 2 4. Find a recurrence relation for the coefficients am+2 in terms of am and m. 5. Use the recurrence relation to find expressions for the coefficients az, as, a4 and as. 3 6. Write down the general solution to (1) in the form y = aof (x) + a19(2). 7. Find the particular solution of (1) that satisfies the initial conditions y(0) = 3 and y'(0) = 2.
Join a community of subject matter experts. Register for FREE to view solutions, replies, and use search function. Request answer by replying!