Thursday, April 20, 2023

Solutions To Power Series Differential Equations Problems

 1. Determine whether x = 0 is an ordinary point of the differential equation:

x2 y " + 2 y'  +  xy = 0 

Soln.

Rewrite in standard form:

y " =   -  2 y' / x2  -   xy  / x2 

y " =   -  (2 / x2) y'-   y  / x   

P(x)  =  2 / x2     

Q (x) =  1/x   

Neither function is analytic at x= 0 so x = 0 is a singular point of the differential equation because the denominators go to 0.

2. Find the power series solution for the differential equation:

xy" +  x3 y   - 3 y = 0   

That satisfies: y = 0 and y' = 2  at x = 1

Soln.:

Rewrite in standard form and differentiate in successive steps:

 y" =  -  x2 y'   +  3 x-1   y

 y"' =  -  x 2 y" - (2x y'  -  3 x-1 ) y'  -   3 x-2 y  

iv  =  -  x 2 y"' - ( 4 x  3 x-1 ) y"  - ( 2 +  6 x-2 )y' +   6 x-3  y

Evaluate these derivatives at x =1 :

y"(1) =  -2  

y"'(1) =  4

y iv (1) = -18 

Then the solution is:

y(x) =  2(x - 1) -  (x - 1) 2   2/3 (x - 1) 3   -  3/4 (x - 1)'   . . . 


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