Fun eode

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lurflurf
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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.
 

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HallsofIvy
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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 xIV= -x.
 

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