【原】transient term及常微分方程2.3节作业

1 transient term 简介

In the context of ordinary differential equations (ODEs), a transient term refers to a term in the general solution of an ODE that decays to zero as time goes to infinity, regardless of the initial conditions.

Transient terms are often contrasted with steady-state or long-term behavior, which refers to the behavior of the solution as time goes to infinity after any transient effects have died out.

For example, in the ODE:

the general solution is:

where and are constants determined by the initial conditions.

In this case, there are no transient terms, since the solution oscillates forever with a constant amplitude and frequency. However, if there were a damping term in the ODE, such as:

then the general solution would include a transient term that decays over time, in addition to the steady-state oscillation:

In this case, the transient term represents the initial decay of any initial perturbations, while the steady-state oscillation represents the long-term behavior of the solution.

确定微分方程的transient term的一种方法是通过分离变量和积分来解决微分方程，并对得到的通解进行分析。

微分方程里的瞬变项（transient term）是指方程的解中与初始条件有关的那部分，也称为自由项（homogeneous solution）。微分方程一般包含两部分：齐次方程和非齐次方程。齐次方程的解是由系统本身的性质决定的，而非齐次方程的解则受到外部因素的影响。瞬变项是非齐次方程的解中与初始条件有关的那部分，它反映了系统在初始状态下的响应，通常是一种短暂的振荡或反应，随着时间的推移逐渐消失，最终达到稳定状态。瞬变项的存在对系统的稳定性、性能和响应产生重要影响，需要进行深入研究和分析。

2. 习题

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution.

The differential equation is:

We can write this in the form:

Separating variables, we get:

Integrating both sides, we get:

where is an arbitrary constant of integration.

Exponentiating both sides, we get:

where is a non-zero constant of integration.

Therefore, the general solution to the differential equation is:

The largest interval over which the general solution is defined is .

The term decays to zero as approaches infinity, so it is a transient term. Therefore, the general solution has a transient term.

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To solve the differential equation , we can use the method of integrating factors. First, we rearrange the equation to get:

The integrating factor is then:

Multiplying both sides of the differential equation by the integrating factor, we get:

This can be written as:

Integrating both sides with respect to , we get:

where is the constant of integration. Solving for , we get the general solution:

The largest interval over which the general solution is defined depends on the value of . Since is always positive, the denominator of the last term in the general solution is never zero. Therefore, the general solution is defined for all .

There are no transient terms in the general solution, since the solution is expressed in terms of a single term that does not decay over .

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To solve the differential equation , we can use the method of integrating factors. First, we rearrange the equation to get:

The integrating factor is then:

Multiplying both sides of the differential equation by the integrating factor, we get:

This can be written as:

Integrating both sides with respect to , we get:

where is the constant of integration. Solving for , we get the general solution:

The largest interval over which the general solution is defined is .

The term decays to zero as approaches infinity, so it is a transient term.

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To solve the differential equation , we can use the method of integrating factors. First, we rearrange the equation to get:

The integrating factor is then:

Multiplying both sides of the differential equation by the integrating factor, we get:

This can be written as:

Integrating both sides with respect to , we get:

where is the constant of integration. Solving for , we get the general solution:

The largest interval over which the general solution is defined for all .

The term is the transient term, which will eventually decay to zero as .

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Solve the given initial-value problem. Give the largest interval over which the solution is defined.

方法一：积分因子法

This is a linear first-order differential equation of the form , where and . The integrating factor is . Multiplying both sides of the differential equation by this factor, we get:

Simplifying the left-hand side using the product rule, we get:

Integrating both sides with respect to , we get:

To find the constants of integration, we use the initial condition . Substituting into the above expression for , we get:

Therefore, the solution is:

where is an arbitrary constant. The interval over which the solution is defined is .

方法二：常数变易法

首先将方程化为标准形式：

考虑齐次方程，

它的通解为， 其中为任意常数。然后考虑非齐次方程，根据常数变易法，设，其中为待定函数。则

将其代入原方程得

化简得

对其积分得

其中为任意常数。因此，原方程的通解为

代入初始条件得，因此特解为

其中.