国外留学,发现高等数学完全不懂了,一定要步骤.Showthattheindicatedfunctiony1(x)isasolutionofthegivendif-ferentialequation.Usereductionoforder,tofindasecondsolutiony2(x).(a)y′′+y=0;y1=sinx(b)xy′′+y′=0;y1

Question

国外留学,发现高等数学完全不懂了,一定要步骤.
Show that the indicated functiony1(x) is a solution of the given dif-
ferential equation.Use reduction of order,to find a second solutiony2(x).
(a)y′′+y=0; y1=sinx
(b)xy′′+y′=0; y1=lnx
(c) (1−2x−x^2)y′′+2(1+x)y′−2y=0; y1=x+1

Answer 1

a. y1'=cosx y1''=-sinx
y1''+y1=-sinx+sinx=0
∴y1(x) is a solution of the given differential equation.
let y2(x)=c(x)y1(x)
y2'=c'y1+cyi'
y2''=c''y1+2c'y1'+cy1''
y2''+y2=c''y1+2c'y1'+cy1''+cy1=c''y1+2c'y1'=0
sinxdc'+2cosxc'dx=0
(sinx)^2dc'+2sinxcosxc'dx=0
d[c'(sinx)^2]=0
c'=c1/(sinx)^2 c=-c1cotx+c2
let c1=-1 c2=0
c=cotx
y2=cy1=cotxsinx=cosx
Similarly, you can calculate the others