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Show that if \(a\) and \(\lambda\) are positive constants, and \(b\) is any real number, then every solution of the equation $$ y^{\prime}+a y=b e^{-\lambda t} $$ has the property that \(y \rightarrow 0\) as \(t \rightarrow \infty\) Hint: Consider the cases \(a=\lambda\) and \(a \neq \lambda\) separately.

Short Answer

Expert verified
Question: For the given first-order linear ordinary differential equation \(y' + ay = be^{-\lambda t}\), show that every solution has the property that \(y \rightarrow 0\) as \(t \rightarrow \infty\). Answer: In both cases (\(a=\lambda\) and \(a\neq\lambda\)), the solutions of the given ODE have the desired property, as shown in the step-by-step solution. Specifically, for \(a = \lambda\), the solution is \(y(t) = \frac{bt + C}{e^{at}}\), and for \(a \neq \lambda\), the solution is \(y(t) = \frac{b}{a-\lambda}e^{-(\lambda - a)t} + De^{-at}\). As \(t \rightarrow \infty\), both expressions approach 0, fulfilling the condition \(y \rightarrow 0\).

Step by step solution

01

Case \(a=\lambda\)

When \(a = \lambda\), our equation becomes: $$ y' + ay = be^{-at} $$ Now, let's solve this first-order linear ODE using an integrating factor. The integrating factor is given by \(I(t) = e^{\int a dt} = e^{at}\).
02

Multiply the equation by the integrating factor

Multiply both sides of the equation by the integrating factor, \(e^{at}\): $$ e^{at}(y' + ay) = be^{at}e^{-at} $$ The left-hand side of the equation now becomes the derivative of the product of \(y(t)\) and the integrating factor, \(e^{at}\): $$ \frac{d}{dt}(e^{at}y) = b $$
03

Integrate both sides of the equation

Now, let's integrate both sides of the equation with respect to \(t\): $$ \int \frac{d}{dt}(e^{at}y) dt = \int b dt $$ On the left-hand side, we can cancel the derivative with the integral, which leaves us with \(e^{at}y\). On the right-hand side, the integral of a constant \(b\) with respect to \(t\) is \(bt + C\), where \(C\) is the integration constant: $$ e^{at}y = bt + C $$
04

Solve for \(y(t)\)

Now, let's solve for \(y(t)\) by dividing both sides by \(e^{at}\): $$ y(t) = \frac{bt + C}{e^{at}} $$ As \(t \rightarrow \infty\), we can see that the denominator \(e^{at}\) grows without bound, so \(y(t) \rightarrow 0\).
05

Case \(a\neq\lambda\)

When \(a \neq \lambda\), our equation remains as it was given: $$ y' + ay = be^{-\lambda t} $$ Again, let's solve this first-order linear ODE using an integrating factor. The integrating factor is still given by \(I(t) = e^{\int a dt} = e^{at}\).
06

Multiply the equation by the integrating factor

Multiply both sides of the equation by the integrating factor, \(e^{at}\): $$ e^{at}(y' + ay) = be^{at}e^{-\lambda t} $$ The left-hand side of the equation now becomes the derivative of the product of \(y(t)\) and the integrating factor, \(e^{at}\): $$ \frac{d}{dt}(e^{at}y) = be^{(a-\lambda)t} $$
07

Integrate both sides of the equation

Next, let's integrate both sides of the equation with respect to \(t\): $$ \int \frac{d}{dt}(e^{at}y) dt = \int be^{(a-\lambda)t} dt $$ On the left-hand side, we can cancel the derivative with the integral, which leaves us with \(e^{at}y\). On the right-hand side, we have: $$ \int be^{(a-\lambda)t} dt = \frac{b}{a-\lambda}e^{(a-\lambda)t} + D $$ where \(D\) is another integration constant.
08

Solve for \(y(t)\)

Now, let's solve for \(y(t)\) by dividing both sides by \(e^{at}\): $$ y(t) = \frac{b}{a-\lambda}e^{(a-\lambda)t}e^{-at} + De^{-at} $$ Rewriting the equation gives: $$ y(t) = \frac{b}{a-\lambda}e^{-(\lambda - a)t} + De^{-at} $$ As \(t \rightarrow \infty\), the term \(e^{-(\lambda - a)t}\) approaches 0, since \(\lambda \neq a\). Similarly, the term \(e^{-at}\) becomes 0 as well. Therefore, we have \(y(t) \rightarrow 0\) in this case as well. In conclusion, for both cases (\(a=\lambda\) and \(a\neq\lambda\)), the solutions of the given ODE have the property \(y \rightarrow 0\) as \(t \rightarrow \infty\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First-Order Linear ODE
A first-order linear ordinary differential equation (ODE) is a type of differential equation that involves a first derivative of a function. In the form \( y' + P(t) y = Q(t) \), it is linear with respect to \( y \). Such equations are common in various scientific fields, providing a tool to model relationships where changes over time are linear. In the exercise, the equation \( y^{\prime}+a y = b e^{-\lambda t} \) fits this pattern.
To solve first-order linear ODEs, we typically use integrating factors or substitution methods. The importance lies in their ability to approximate real-world phenomena such as electrical circuits, population growth, or decay processes. The linearity implies proportionality and simplicity but may not capture all complex dynamics, being effective within certain bounds of initial conditions and linear behaviors.
Integrating Factor
An integrating factor is a function used to solve linear differential equations more easily. It multiplies every term in the equation, simplifying it into a format that integrates cleanly. For a given differential equation \( y' + P(t) y = Q(t) \), the integrating factor is given by \( \mu(t) = e^{\int P(t) dt} \).
  • The goal is to transform the left-hand side of the equation into a derivative of a product.
  • In the exercise provided, the integrating factor \( I(t) = e^{at} \) allows for the equation to be rewritten as the derivative \( \frac{d}{dt}(e^{at}y) \).
By solving the resulting equation, integration becomes straightforward, leading to expressions where continuous and differentiable functions can be solved in terms of elementary functions, often resulting in straightforward expressions for the solution \( y(t) \). The clarity of this process highlights why integrating factors are preferred in solving linear ODEs.
Limit as \(t \rightarrow \infty\)
Understanding the behavior of solutions as time \( t \) moves towards infinity helps us foresee long-term trends in models and equations. The asymptotic behavior analyzed, as shown in the step-by-step solution, determines the ultimate disposition of functions in time.
  • In our exercise, regardless of whether \( a = \lambda \) or \( a eq \lambda \), the solution \( y(t) \) approaches \( 0 \) as \( t \rightarrow \infty \).
  • It is a crucial property for confirming if a model naturally decomposes over time or if it will stabilize to a non-zero state.
Such analyses are vital in areas like physics (for understanding damped motion), biological systems (for determining long-term rates of reaction), and economics (for evaluating the effect of decay rates on investments). Mathematical rigor in proving these limits helps ensure models' reliability and applicability in forecasting scenarios.
Exponential Decay
Exponential decay describes processes where the quantity reduces at a rate proportional to its current value. It is one of the key results in analyzing differential equations where coefficients remain constant.
  • In our exercise, when computing \( y(t) \, ext{as} \, t \rightarrow \infty \), terms like \( e^{at} \) and \( e^{-(\lambda - a)t} \) emerge.
  • These terms decay exponentially if the exponent is negative, steadily driving \( y(t) \) towards zero over time.
The phenomenon is frequently encountered in physics, such as radioactive decay or thermal cooling and is adjustable to encompass more complex processes, modifying assumptions to fit various decay or growth rates. It highlights the practical, intuitive understandings in mathematics, emphasizing how initial parameters critically determine time evolution and eventual equilibrium states inside controlled systems.

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Most popular questions from this chapter

Epidemics. The use of mathematical methods to study the spread of contagious diseases goes back at least to some work by Daniel Bernoulli in 1760 on smallpox. In more recent years many mathematical models have been proposed and studied for many different diseases. Deal with a few of the simpler models and the conclusions that can be drawn from them. Similar models have also been used to describe the spread of rumors and of consumer products. Suppose that a given population can be divided into two parts: those who have a given disease and can infect others, and those who do not have it but are susceptible. Let \(x\) be the proportion of susceptible individuals and \(y\) the proportion of infectious individuals; then \(x+y=1 .\) Assume that the disease spreads by contact between sick and well members of the population, and that the rate of spread \(d y / d t\) is proportional to the number of such contacts. Further, assume that members of both groups move about freely among each other, so the number of contacts is proportional to the product of \(x\) and \(y .\) since \(x=1-y\) we obtain the initial value problem $$ d y / d t=\alpha y(1-y), \quad y(0)=y_{0} $$ where \(\alpha\) is a positive proportionality factor, and \(y_{0}\) is the initial proportion of infectious individuals. (a) Find the equilibrium points for the differential equation (i) and determine whether each is asymptotically stable, semistable, or unstable. (b) Solve the initial value problem (i) and verify that the conclusions you reached in part (a) are correct. Show that \(y(t) \rightarrow 1\) as \(t \rightarrow \infty,\) which means that ultimately the disease spreads through the entire population.

Consider the initial value problem \(y^{\prime}=y^{1 / 3}, y(0)=0\) from Example 3 in the text. (a) Is there a solution that passes through the point \((1,1) ?\) If so, find it. (b) Is there a solution that passes through the point \((2,1)\) ? If so, find it. (c) Consider all possible solutions of the given initial value problem. Determine the set of values that these solutions have at \(t=2\)

Show that if \(\left(N_{x}-M_{y}\right) / M=Q,\) where \(Q\) is a function of \(y\) only, then the differential equation $$ M+N y^{\prime}=0 $$ has an integrating factor of the form $$ \mu(y)=\exp \int Q(y) d y $$

Suppose that a certain population satisfies the initial value problem \(d y / d t=r(t) y-k, \quad y(0)=y_{0}\) where the growth rate \(r(t)\) is given by \(r(t)=(1+\sin t) / 5\) and \(k\) represents the rate of predation. $$ \begin{array}{l}{\text { (a) Supposs that } k=1 / 5 \text { . Plot } y \text { versust for several values of } y_{0} \text { between } 1 / 2 \text { and } 1 \text { . }} \\ {\text { (b) P. Stimate the critical initial population } y_{e} \text { below which the population will become }} \\ {\text { extinct. }} \\ {\text { (c) Choose other values of } k \text { and find the correponding } y_{i} \text { for each one. }} \\ {\text { (d) Use the data you have found in parts }(a) \text { and }(b) \text { to plot } y_{c} \text { versus } k \text { . }}\end{array} $$

Determine whether or not each of the equations is exact. If it is exact, find the solution. $$ (x \ln y+x y) d x+(y \ln x+x y) d y=0 ; \quad x>0, \quad y>0 $$

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