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
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