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Chapter 9: Problem 22

What integral equation is equivalent to the initial value problem \(y^{\prime}=f(x), y\left(x_{0}\right)=y_{0} ?\)

Short Answer

Expert verified
\( y(x) = y_{0} + \int_{x_{0}}^{x} f(t) \, dt \) is the equivalent integral equation.

Step by step solution

01

Understand the Initial Value Problem

The given initial value problem consists of a first-order differential equation, \( y^{\prime} = f(x) \), and an initial condition \( y(x_{0}) = y_{0} \). This tells us that the rate of change of \( y \) with respect to \( x \) is specified by the function \( f(x) \), and at the point \( x = x_{0} \), the value of \( y \) is initially \( y_{0} \).
02

Recognize the Need for an Integral Equation

We need to express the given differential equation as an equivalent integral equation. An integral equation rewrites the derivative \( y^{\prime} = f(x) \) in terms of \( y \) as an integration accumulated over an interval starting from \( x_{0} \).
03

Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, if \( y^{\prime} = f(x) \), then \( y(x) \) can be expressed by integrating \( f(x) \) from \( x_{0} \) to \( x \) and adding the initial value \( y(x_{0}) = y_{0} \). Therefore, the equivalent integral equation is:\[ y(x) = y_{0} + \int_{x_{0}}^{x} f(t) \, dt \]
04

Finalize the Integral Equation

The integral equation \( y(x) = y_{0} + \int_{x_{0}}^{x} f(t) \, dt \) expresses \( y(x) \) as the sum of the initial value \( y_{0} \) and the accumulated area under the curve \( f(t) \) from \( x_{0} \) to \( x \). This equation satisfies both conditions of the initial value problem.

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Key Concepts

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

Initial Value Problem
An initial value problem is a type of differential equation that provides not only a differential equation but also an initial condition. Think of it as a story with a beginning. You know where you start, and now you have to figure out where you'll end up as things change.

In mathematical terms, it usually involves:
  • A differential equation, such as \( y' = f(x) \), which tells us how the function \( y \) changes with respect to \( x \).
  • An initial condition, for example, \( y(x_0) = y_0 \), which gives us a specific value for \( y \) when \( x = x_0 \).
The beauty of solving an initial value problem lies in its ability to predict future behavior based on current conditions. By solving it, we determine a function \( y(x) \) that satisfies both the differential equation and the initial condition. This process is crucial in fields like physics and engineering.
Differential Equation
A differential equation describes relationships involving the rates at which things change. Essentially, it is an equation that involves an unknown function and its derivatives. These equations can be thought of as the mathematical language of change.

They come in various forms:
  • Ordinary Differential Equations (ODEs): Involve functions of a single variable and their derivatives, like \( y' = f(x) \).
  • Partial Differential Equations (PDEs): Involve multiple variables and their partial derivatives, common in physics.
Differential equations are powerful tools. They are used extensively across sciences and engineering to model the behavior of complex systems. For instance, understanding how populations grow or how heat diffuses through a solid can be described by differential equations. Thus, they help us predict and understand the dynamic changes within a system.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links the worlds of differentiation and integration, bridging two core concepts in calculus. It provides a way to evaluate definite integrals by using antiderivatives.

Here's what it tells us:
  • If a function \( f \) is continuous over a closed interval \([a, b]\), and \( F \) is an antiderivative of \( f \), then the integral of \( f \) from \( a \) to \( b \) can be computed as \( F(b) - F(a) \).
  • It shows that differentiation and integration are inverse processes.
In the context of solving an initial value problem, the theorem is applied to convert a differential equation into an integral equation. When we know the rate of change \( f(x) \), we can integrate it to find the total change over an interval, adding any initial conditions to find the specific solution we're interested in. This makes it a powerful tool in solving a wide array of problems.

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Most popular questions from this chapter

Use a CAS to explore graphically each of the differential equations.Perform the following steps to help with your explorations. a. Plot a slope field for the differential equation in the given \(x y\) -window. b. Find the general solution of the differential equation using your CAS DE solver. c. Graph the solutions for the values of the arbitrary constant \(C=-2,-1,0,1,2\) superimposed on your slope field plot. d. Find and graph the solution that satisfies the specified initial condition over the interval \([0, b].\) e. Find the Euler numerical approximation to the solution of the initial value problem with 4 subintervals of the \(x\) -interval and plot the Euler approximation superimposed on the graph produced in part (d). f. Repeat part (e) for \(8,16,\) and 32 subintervals. Plot these three Euler approximations superimposed on the graph from part (e). g. Find the error ( \(y\) (exact) \(-y\) (Euler)) at the specified point \(x=b\) for each of your four Euler approximations. Discuss the improvement in the percentage error. $$y^{\prime}=y(2-y), \quad y(0)=1 / 2 ; \quad 0 \leq x \leq 4,0 \leq y \leq 3 ; b=3.$$

The fish and game department in a certain state is planning to issue hunting permits to control the deer population (one deer per permit). It is known that if the deer population falls below a certain level \(m,\) the deer will become extinct. It is also known that if the deer population rises above the carrying capacity \(M,\) the population will decrease back to \(M\) through disease and malnutrition. a. Discuss the reasonableness of the following model for the growth rate of the deer population as a function of time: $$\frac{d P}{d t}=r P(M-P)(P-m),$$ where \(P\) is the population of the deer and \(r\) is a positive constant of proportionality. Include a phase line. b. Explain how this model differs from the logistic model \(d P / d t=r P(M-P) .\) Is it better or worse than the logistic model? c. Show that if \(P > M\) for all \(t,\) then \(\lim _{t \rightarrow \infty} P(t)=M\). d. What happens if \(P < m\) for all \(t ?\) e. Discuss the solutions to the differential equation. What are the equilibrium points of the model? Explain the dependence of the steady-state value of \(P\) on the initial values of \(P .\) About how many permits should be issued?

Suppose that a pearl is sinking in a thick fluid, like shampoo, subject to a frictional force opposing its fall and proportional to its velocity. Suppose that there is also a resistive buoyant force exerted by the shampoo. According to Archimedes' principle, the buoyant force equals the weight of the fluid displaced by the pearl. Using \(m\) for the mass of the pearl and \(P\) for the mass of the shampoo displaced by the pearl as it descends, complete the following steps. a. Draw a schematic diagram showing the forces acting on the pearl as it sinks, as in Figure 9.19. b. Using \(v(t)\) for the pearl's velocity as a function of time \(t\) write a differential equation modeling the velocity of the pearl as a falling body. c. Construct a phase line displaying the signs of \(v^{\prime}\) and \(v^{\prime \prime}\). d. Sketch typical solution curves. e. What is the terminal velocity of the pearl?

Solve the initial value problems. \(\frac{d y}{d x}+x y=x, \quad y(0)=-6\)

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places. $$y^{\prime}=y^{2}(1+2 x), \quad y(-1)=1, \quad d x=0.5$$
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