Consider the initial value problem ẋ=y / z, ẏ=-x / z (x, y, z)=(1,0,1) at t=1 .

Question

Consider the initial value problem $$ \begin{aligned} & \dot{x}=y / z, \\ & \dot{y}=-x / z \quad(x, y, z)=(1,0,1) \text { at } t=1 . \\ & \dot{z}=1, \end{aligned} $$ (a) Convert this system to cylindrical coordinates $(r, \theta, \zeta)$, where $r^2=x^2+y^2$, $\zeta=z$, and $\theta=\arctan (y / x)$. Find the initial conditions in the new coordinate system. (b) Solve the new system and show that its solution exists in the maximal interval $J=(0, \infty)$. (c) Apply Theorem 3.10 to the new system and determine the maximal interval guaranteed by the theorem.

   Consider the initial value problem
$$
\begin{aligned}
& \dot{x}=y / z, \\
& \dot{y}=-x / z \quad(x, y, z)=(1,0,1) \text { at } t=1 . \\
& \dot{z}=1,
\end{aligned}
$$
(a) Convert this system to cylindrical coordinates $(r, \theta, \zeta)$, where $r^2=x^2+y^2$, $\zeta=z$, and $\theta=\arctan (y / x)$. Find the initial conditions in the new coordinate system.
(b) Solve the new system and show that its solution exists in the maximal interval $J=(0, \infty)$.
(c) Apply Theorem 3.10 to the new system and determine the maximal interval guaranteed by the theorem.

Differential Dynamical Systems

James D. Meiss 1st Edition

Chapter 3, Problem 12

Instant Answer

Step 1

We start with the given system of equations: \[ \begin{aligned} & \dot{x} = \frac{y}{z}, \\ & \dot{y} = -\frac{x}{z}, \\ & \dot{z} = 1. \end{aligned} \] We define the cylindrical coordinates as follows: - $ r = \sqrt{x^2 + y^2} $ - \( \theta = Show more…

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Consider the initial value problem $$ \begin{aligned} & \dot{x}=y / z, \\ & \dot{y}=-x / z \quad(x, y, z)=(1,0,1) \text { at } t=1 . \\ & \dot{z}=1, \end{aligned} $$ (a) Convert this system to cylindrical coordinates $(r, \theta, \zeta)$, where $r^2=x^2+y^2$, $\zeta=z$, and $\theta=\arctan (y / x)$. Find the initial conditions in the new coordinate system. (b) Solve the new system and show that its solution exists in the maximal interval $J=(0, \infty)$. (c) Apply Theorem 3.10 to the new system and determine the maximal interval guaranteed by the theorem.

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