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  

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)'   . . .