Chapter 7.3, Problem 25E

(a)

To determine

To show that ${\left(x^2+y^2+z^2\right)}^{\prime }=0$ if $x^2+y^2+z^2=1$.

(b)

To determine

To show: The solution with initial point $\left(x_0,y_0,z_0\right)$ remains in the plane $z=z_0$.

(c)

To determine

To show: The limit cycle for the solution with initial point $\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$ is the circle $x^2+y^2=\frac{3}{4}$ in the plane $z=\frac{1}{2}$.