Ein Schimmelpilz wächst mit einer Geschwindigkeit, die proportional zur vorhandenen Menge ist. Innerhalb von 24 Stunden ist die Menge von 2 Gramm auf 3 Gramm angewachsen. Wie viele Gramm davon sind nach weiteren 24 Stunden vorhanden?

The solution you found in the indicated exercise depends on the parameters k and b. Describe the qualitative behavior of each solution in the following situations in terms of the motion of the "mass-spring with handle" model in this section (assume $\omega \approx 1$):

(a) k large, b and $\omega$ small but positive.

(b) b large, k and $\omega$ small but positive.

In the following exercise,

(a) find the general solution of the given differential equation, and

(b) compute the amplitude and phase angle of the steady-state solution.

$$\frac{d^2 y}{d t^2}+4 \frac{d y}{d t}+3 y=5 \sin 2 t$$

Solve the initial-value problem

$$\begin{aligned} & \frac{d x}{d t}=3 x \\ & \frac{d y}{d t}=x-2 y, \end{aligned}$$

where the initial condition $(x(0), y(0))$ is:

(2, 2)

For the system

$$\begin{aligned} & \frac{d x}{d t}=-x \sin y+2 y \\ & \frac{d y}{d t}=-\cos y \end{aligned}$$

(a) check that the system is a Hamiltonian system with Hamiltonian function

$$H(x, y)=x \cos y+y^2$$

(b) sketch the level sets of H, and

(c) sketch the phase portrait of the system. Include a description of all equilibrium points and any saddle connections.

In this section we computed a particular solution of the equation

$$\frac{d^2 y}{d t^2}+p \frac{d y}{d t}+q y=\cos \omega t$$

of the form

$$y_p(t)=A \cos (\omega t+\phi)$$

where the phase angle $\phi$ satisfies the equation

$$\tan \phi=\frac{-p \omega}{q-\omega^2}$$

and $-180^{\circ}<\phi<0$. The angle $\phi$ is a function $\phi(\omega, p, q)$ of the parameters $\omega, p$, and $q$.

Compute $\partial^2 \phi / \partial \omega^2$.

If a nonlinear system depends on a parameter, then the equilibrium points can change as the parameter varies. In other words, as the parameter changes, a bifurcation can occur. Consider the one-parameter system family of systems

$$\begin{aligned} & \frac{d x}{d t}=x^2-a \\ & \frac{d y}{d t}=-y\left(x^2+1\right) \end{aligned}$$

where a is the parameter.

(a) Show that the system has no equilibrium points if a < 0.

(b) Show that the system has two equilibrium points if a > 0.

(c) Show that the system has exactly one equilibrium point if a = 0.

(d) Find the linearization of the equilibrium point for a = 0 and compute the eigenvalues of this linear system.

Remark: The system changes from having no equilibrium points to having two equilibrium points as the parameter a is increased through a = 0. We say that the system has a bifurcation at a = 0, and that a = 0 is a bifurcation value of the parameter.

주어진 초기 값 문제가 고유한 이중 미분 가능한 해를 가질 것이 확실한 가장 긴 간격을 결정하라.해결책을 찾으려 하지 마십시오.

t(t−4)y”+3ty'+4y=2,y(3)=0,y'(3)=−1

Continuing the study of the nonlinear system given in the indicated exercise,

(a) use the direction field to sketch the phase portrait for the system if a = -1,

(b) use the direction field and the linearization at the equilibrium point to sketch the phase portrait for a = 0, and

(c) use the direction field and the linearization at the equilibrium points to sketch the phase portrait for a = 1.

Remark: The transition from a system with no equilibrium points to a system with one saddle and one sink via a system with one equilibrium point with zero as an eigenvalue is typical of bifurcations that create equilibria.

Consider the differential equation given below and fix q so that the amplitude of the $\omega_1$-term is greatest.

$$\frac{d^2 y}{d t^2}+p \frac{d y}{d t}+q y=\cos \omega_1 t+\cos \omega_2 t .$$

How does this ratio depend on p ? (For example, what happens as $p \rightarrow 0$ and $p \rightarrow \infty$, etc.?)

trovare gli autovalori e le autofunzioni del problema ai limiti dato. Supponiamo che tutti gli autovalori siano reali. $y''+λy=0,y'[(0)=0,y(π)=0$

In the following exercise, each system depends on the parameter a. In the given exercise,

(a) find all equilibrium points,

(b) determine all values of a at which a bifurcation occurs, and

(c) in a short paragraph complete with pictures, describe the phase portrait at, before, and after each bifurcation value.

$$\begin{aligned} & \frac{d x}{d t}=y-x^2 \\ & \frac{d y}{d t}=a \end{aligned}$$

In the following predator-prey population models, $x$ represents the prey, and $y$ represents the predators. (i) $\frac{d x}{d t}=5 x-3 x y$

$\quad \frac{d y}{d t}=-2 y+6 x y$

(ii) $\frac{d x}{d t}=x-8 x y$

$\quad \frac{d y}{d t}=-2 y+\frac{1}{2} x y$

(a) In which system does the prey reproduce more quickly when there are no predators (when $y=0$ ) and equal numbers of prey? (b) In which system are the predators more successful at catching prey? In other words, if the number of predators and prey are equal for the two systems, in which system do the predators have a greater effect on the rate of change of the prey? (c) Which system requires more prey for the predators to achieve a given growth rate (assuming identical numbers of predators in both cases)?

Suppose that heat is generated within a laterally insulated rod at the rate of $q ( x , t )$ calories per second per cubic centimeter. Extend the derivation of the heat equation in this section to derive the equation

$$\frac { \partial u } { \partial t } = k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { q ( x , t ) } { c \delta }$$

Consider a unit mass sliding on a frictionless table attached to a spring, with spring constant k = 16. Suppose the mass is lightly tapped by a hammer every T seconds.

Suppose that the first tap occurs at time t = 0 and before that time the mass is at rest. Describe what happens to the motion of the mass for the following choices of the tapping period T:

(a) T = 1

(b) T = 3/2

(c) T = 2

(d) T = 5/2

(e) T = 3