- the form
x 2 − n y 2 = 1 , {\displaystyle
x^{2}-ny^{2}=1,}
where n is a
given positive nonsquare integer, and
integer solutions are
sought for
x and y....
-
proved that a Thue
equation can have only
finitely many
solutions in
integers x {\displaystyle
x} and y {\displaystyle y} , a
result known as Thue's theorem...
-
service situations where the
question is
about an end user's
attempted solution (
X)
rather than the root
problem itself (Y or Why?). The XY
problem obscures...
-
solution (
x ,
x 1 , … ,
x k ) {\displaystyle (
x,
x_{1},\dots ,
x_{k})} such that 0 ≤
x < N, and that the
other solutions are
obtained by
adding to
x a...
- {\displaystyle
x_{e}} is
Lyapunov stable and all
solutions that
start out near
x e {\displaystyle
x_{e}}
converge to
x e {\displaystyle
x_{e}} , then
x e {\displaystyle...
-
x −
x 1 ) (
x −
x 2 ) =
x 2 − (
x 1 +
x 2 )
x +
x 1
x 2 = 0 {\displaystyle \left(
x-
x_{1}\right)\left(
x-
x_{2}\right)=
x^{2}-\left(
x_{1}+
x_{2}\right)
x+
x_{1}x_{2}=0}...
- satisfied. In the
example above, a
solution is
given by the
ordered triple (
x , y , z ) = ( 1 , − 2 , − 2 ) , {\displaystyle (
x,y,z)=(1,-2,-2),}
since it makes...
- The
solution set of the
single equation x = 0 {\displaystyle
x=0} is the set { 0 } {\displaystyle \{0\}} .
Since there do not
exist numbers x {\displaystyle...
- all
values of x {\displaystyle
x} , so the
solution set is all real numbers. But
clearly not all real
numbers are
solutions to the
original equation. The...
- A
singular solution ys(
x) of an
ordinary differential equation is a
solution that is
singular or one for
which the
initial value problem (also called...
