Eg. Solve by series expansions. y’+ky=0 (Sol.) y=, , 1, 2, 3, , n=0, … Eg. Solve (x-1)y”-xy’+y=0. (Ans.) y(x)=c1ex+ c2x Eg. Solve (x+1)y”-(x+2)y’+y=0. (Ans.) y(x)= c1ex+ c2(x+2) Eg. Solve by series expansions. y”+k2y=0 (Sol.) y=, ; Eg. Solve by series expansions. y”-exy=0 (Sol.) y=, , , , , , … Eg. Solve y”-exy=sin(x)+1. 3-2 Method of Frobenius For If and have a regular singular point at and are P(x) y”+Q(x)y’+R(x)y=0. x0=0, analytical at then is a solutionr2+Ar+B=0r=r1, x0=0, y= r2 and is not an integer, then Case 1 r1≠r2 r1-r2 and is a positive integer, then and , where may be 0. Case 2 r1-r2 A then and Case 3 r1=r2, Eg. Solve 3xy”+y’-y=0. (Sol.) , and - have a regular singular point at 0, but and - are analytic at 0. , , ,
Eg. Solve y’+ky=0 by series expansions. (Sol.) y=, , n=0, 1, 2, 3, … , Eg. Solve (x-1)y”-xy’+y=0. (Ans.) y(x)=c1ex+ c2x Eg. Solve (x+1)y”-(x+2)y’+y=0. (Ans.) y(x)= c1ex+ c2(x+2) Eg. Solve y”+k2y=0 by series expansions. (Sol.) y=, ; Eg. Solve y”-exy=0 by series expansions. (Sol.) y=, , , , , , … Eg. Solve y”-exy=sin(x)+1. 3-2 Method of Frobenius For P(x) y”+Q(x)y’+R(x)y=0. If and have a regular singular point at x0=0, and are analytical at x0=0, then y= is a solutionr2+Ar+B=0r=r1, r2 Case 1 r1≠r2 and r1-r2 is not an integer, then and Case 2 r1-r2 is a positive integer, then and , where A may be 0. Case 3 r1=r2, then and Eg. Solve 3xy”+y’-y=0. (Sol.) , and - have a regular singular point at 0, but and - are analytic at 0. , , ,
2016Math3.docx