AboutAbe Mantell Expertise Hello, I am a college professor of mathematics and regularly teach all levels from elementary mathematics through differential equations, and would be happy to assist anyone with such questions!
Experience Over 15 years teaching at the college level.
Question 1.Determine the singular points of the given differential equation, then classify each singular point as regular or irregular.
(x^2)(x+1)y'' + ((x^2)-1)y' - y = 0
2. Solve xy'' - xy' + y = 0
3. x=0 is a regular singular point of the differential equation:
(x^2 + x - 6)y'' - (x+3)y' + (x-2)y = 0,
Determine the form of the solution of the differential equation.
Answer 1. Re-written as: y'' + (x^2-1)/(x^2(x+1)) y' - y/(x^2(x+1)) = 0
. Singular points where x^2(x+1)=0...thus x=0 and x=-1 are singular pts.
. Since lim (as x->0) x[(x^2-1)/(x^2(x+1))] DNE, x=0 is an irregular
. singular pt. Since lim (as x->-1) (x+1)[(x^2-1)/(x^2(x+1))]=0
. and lim (as x->-1) (x+1)^2[(x^2-1)/(x^2(x+1))]=0, x=-1 is a regular
. singular pt.
2. The solution works out to be:
. y(x)=C1*x+C2*(-x*ln(x)+1+x-(1/2)*x^2-(1/12)*x^3-(1/72)*x^4-
(1/480)*x^5-(1/3600)*x^6-(1/30240)*x^7-(1/282240)*x^8+...)
3. The solutions works out to be:
. y(x) = y(0)+y'(0)*x+(-(1/4)*y'(0)-(1/6)*y(0))*x^2+(-(1/18)*y'(0)+
(5/108)*y(0))*x^3+((5/216)*y'(0)+(7/2592)*y(0))*x^4+(-(11/4320)*y'(0)
-(13/12960)*y(0))*x^5+((61/155520)*y'(0)+(149/466560)*y(0))*x^6+
(-(17/816480)*y'(0)-(253/9797760)*y(0))*x^7+((439/26127360)*y'(0)+
(2557/156764160)*y(0))*x^8+...
