Question

国外留学，发现高等数学完全不懂了，求大神速度帮忙解决3题，万分感谢。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

Answer 2

These differential equations have a common characteristic. They all have a same solution which is 0.
So actually u can add a modulus in the front.
which means if the Y1=sinx, Y2 can be ksinx, such as 2sinx, 3sinx.

It's very easy, hope u understand.

Answer 3

你把要求翻译一下，最烦看英文了。。。。。。

Answer 4

不是原题 题目最后自己写清楚