While I was studying the Riccati Equations I encountered the following one:
x^2*(y'(x)+y^2)=2(x*y(x)-1).
The basic idea is two guess an appropriate a function y(x) such that you can find 2 appropriate solutions with which you can find y(x) using the formula:
(y-y1 )/(y-y2 )=C*eintegral of b*(y1+y2)dx (1)
where b(x) represents the coefficient of y^2 in the Riccati equation:y'+a(x)*y=b(x)*y^2+c(x);
We solved something similar with this problem.
There was y'+sinx*y^2=2*(sinx/cos2 x)
In this case we took b=-sinx and y=p/cosx where p represents a constant.
We replaced y with p/cosx and since all terms of this equation are multiplied with sinx/cos2 x we solved the 2nd order equation p2 +p-2=0.From here we get p1 =1 and p2 =-2.Now we used the formula (1) and get the result.
So having in mind this type of solving method i was thinking that in my problem i coud take y=p/x and b=x2 .After making all the replacements I got p2 -3p+2=0.Which is nice since i get p=2 and p=1.And from here everything is relatively simple.
I just wantd to ask if these are the right choices.Do you have better ideas?Or,I don't know,different methods to solve this equation?
x^2*(y'(x)+y^2)=2(x*y(x)-1).
The basic idea is two guess an appropriate a function y(x) such that you can find 2 appropriate solutions with which you can find y(x) using the formula:
(y-y1 )/(y-y2 )=C*eintegral of b*(y1+y2)dx (1)
where b(x) represents the coefficient of y^2 in the Riccati equation:y'+a(x)*y=b(x)*y^2+c(x);
We solved something similar with this problem.
There was y'+sinx*y^2=2*(sinx/cos2 x)
In this case we took b=-sinx and y=p/cosx where p represents a constant.
We replaced y with p/cosx and since all terms of this equation are multiplied with sinx/cos2 x we solved the 2nd order equation p2 +p-2=0.From here we get p1 =1 and p2 =-2.Now we used the formula (1) and get the result.
So having in mind this type of solving method i was thinking that in my problem i coud take y=p/x and b=x2 .After making all the replacements I got p2 -3p+2=0.Which is nice since i get p=2 and p=1.And from here everything is relatively simple.
I just wantd to ask if these are the right choices.Do you have better ideas?Or,I don't know,different methods to solve this equation?