Question

In each exercise, discuss the behavior of the solution \(y(t)\) as \(t\) becomes large. Does \(\lim _{t \rightarrow \infty} y(t)\) exist? If so, what is the limit? \(y^{\prime}+y+y \cos t=1+\cos t, \quad y(0)=3\)

Short answer

#Answer# The limit of the solution \(y(t)\) as \(t\) goes to infinity exists and is equal to zero.

Step-by-step solution

Identify the differential equation components

The given differential equation is \(y^{\prime}+y+y\cos t=1+\cos t.\) It is first-order, linear, and non-homogeneous with an initial condition \(y(0)=3\).

Rewrite the equation in standard form

To solve the differential equation, first rewrite it in standard form: \(y^{\prime}+y(1+\cos t)=1+\cos t.\)

Identify the integrating factor

The integrating factor is \(\mu(t)=e^{\int(1+\cos t)dt}\). We can use integration by parts to compute the integral: \(\int(1+\cos t)dt=t+\int\cos tdt=t+\sin t+C\). Thus, the integrating factor is \(\mu(t)=e^{t+\sin t}\).

Multiply the equation by the integrating factor

Multiplying our differential equation by the integrating factor, we obtain: \(e^{t+\sin t}y^{\prime}+(1+\cos t)e^{t+\sin t}y=e^{t+\sin t}(1+\cos t)\)

Recognize the equation as a product rule

Now, notice that the left-hand side of the equation represents the derivative of the product of \(y(t)\) and our integrating factor, \(\mu(t)\): \(\frac{d}{dt}[y(t)e^{t+\sin t}]=e^{t+\sin t}(1+\cos t)\)

Integrate both sides

Integrate both sides of the equation with respect to \(t\): \(y(t)e^{t+\sin t}=\int e^{t+\sin t}(1+\cos t)dt + C\)

Solve for \(y(t)\)

To determine \(y(t)\), divide by the integrating factor, \(e^{t+\sin t}\), on both sides: \(y(t)=\frac{\int e^{t+\sin t}(1+\cos t)dt + C}{e^{t+\sin t}}\)

Apply the initial condition

Now, we will apply the initial condition \(y(0)=3\). Plugging \(t=0\) into our expression for \(y(t)\), we get: \(3=\frac{\int e^{\sin t}(1+\cos t)dt + C}{1}\) This allows us to determine the value of the constant \(C\) and thus the general solution for \(y(t)\).

Analyze the solution as \(t\rightarrow \infty\)

To analyze the behavior of the solution as \(t \rightarrow \infty\), observe that the exponential term \(e^{t+\sin t}\) in the denominator behaves as \(e^t\) for large values of \(t\). Since the exponential growth will always dominate any polynomial growth, the overall expression \(y(t)\) approaches zero as \(t \rightarrow \infty\). Therefore, the limit exists and is equal to zero: \(\lim_{t\rightarrow \infty} y(t)=0\)

Key concepts

First-order Linear Differential Equation

A first-order linear differential equation is an equation involving a function and its first derivative. The general form is given by:\[ \frac{dy}{dt} + P(t)y = Q(t) \]This type of equation is called "linear" because it consists of terms that are either constants or products of constant coefficients and the dependent variable, without exponents other than one.

• "First-order" specifies that the highest derivative in the equation is the first derivative.
• This can demonstrate how systems change over time or space, like decay or population models.

By understanding this, we can apply specific solving techniques for linear differential equations. In our exercise, the equation is first-order because it only involves \(y'\) and \(y\). We typically address these equations using methods like the integrating factor.

Non-homogeneous Equation

A non-homogeneous differential equation is one where the terms do not align to zero on one side of the equation, usually because of an independent function term. In our example:\[ y' + y + y\cos(t) = 1 + \cos(t) \]The presence of \(1 + \cos(t)\) makes the equation non-homogeneous.

• "Non-homogeneous" indicates the equation has an external forcing factor.
• These equations predict behavior beyond natural responses, influenced by external factors.

Understanding this distinction helps define the appropriate method for finding particular solutions compared to its homogeneous counterparts, where solutions derive from natural system responses alone. In practice, solving a non-homogeneous equation involves addressing both the homogeneous and particular solutions.

Integrating Factor

An integrating factor is a function used to simplify solving a linear differential equation. It's specifically useful in transforming the equation into a more manageable form, often simplifying the derivatives into a solvable product:\[ \mu(t)y' + \mu(t)Py = \mu(t)Q \]In our example, the integrating factor is calculated as:\[ \mu(t) = e^{\int (1 + \cos(t))dt } = e^{t + \sin(t)} \]

• By multiplying the entire equation with this factor, the left-hand side can become the derivative of \(\mu(t)y\).
• This allows the problem to reduce into integrating simple terms.

Ultimately, using the integrating factor is a strategic step in solving differential equations by bringing them into a form directly integrable.

Limits and Asymptotic Behavior

Analyzing limits and asymptotic behavior helps to understand how solutions behave as variables approach infinity or specific points. For the given equation, the focus is on understanding:\[ \lim_{t \rightarrow \infty} y(t) \]In practice, as \(t\) grows large, we observe the influence of exponential terms:

• The term \(e^{t+\sin(t)}\) governs the behavior due to its rapid growth.
• The presence of an exponential in the denominator suggests decay in the solution \(y(t)\).

Thus, it is found that as \(t\) becomes very large, the solution approaches zero. This reflects the solution's asymptotic behavior, where for large \(t\), the solution stabilizes to a fixed target or trend. Here, \(y(t)\) approaches zero, making it easier to gauge future system behavior based on its mathematical model.