ry y I SolutionLet x0y0 be a point on the line y xThen we

Solution

Let (x_0,y₀) be a point on the line y = x.Then we must have y₀ = x₀ and so the vector defined by (x^.y^.) is (x₀,x₀), which indeed lies on the line y = x,hence the line y = x is invariant.

Let U(t) =x(t ) - y(t)

Then we have u ^.= x ^. - y^.

   ^{= y - x}

   ^{= - (x - y)}

   ^{= -u}

^{solving the differential equation u.=-u .since u(t) converges to zero as t tends to infinity we indeed have I x(t)-y(t) Iconverges to zero as t tends to infinity.}