Solving an Ordinary Differential Equation Problem: x*x''-y*y''=0

[lurflurf]

I have been looking though Ince (Ordinary Differential Equations), nice book.
I enjoyed solving this problem:
6) pp. 157
Integrate the system
x*x''-y*y''=0
x''+y''+x+y=0
[Edinburgh, 1909.]
we may assume x and y are twice differentiable functions
defined everywhere from reals to reals
' denotes differentiation

I would be interested in the methods used by others.

 

[HallsofIvy]

I have been looking though Ince (Ordinary Differential Equations), nice book.
I enjoyed solving this problem:
6) pp. 157
Integrate the system
x*x''-y*y''=0
x''+y''+x+y=0
[Edinburgh, 1909.]
we may assume x and y are twice differentiable functions
defined everywhere from reals to reals
' denotes differentiation

I would be interested in the methods used by others.

Well, my first reaction would be that yy"= xx" so x"= (y/x)y".
Then x"+ y"+ x+ y= 0 becomes (y/x)y"+ y"= (y+x)y"/x= -(x+y) so y"= -x. Put that back into the original equation and we have x"= y. Differentiating twice more gives x^IV= -x.