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Chapter 8: Problem 25

For the following initial value problems, compute the first two approximations \(u_{1}\) and \(u_{2}\) given by Euler's method using the given time step. $$y^{\prime}(t)=2 y, y(0)=2 ; \Delta t=0.5$$

Short Answer

Expert verified
Answer: The first two approximations are \(u_1 = 4\) and \(u_2 = 8\).

Step by step solution

01

Identify the given problem and initial condition

We have the initial value problem \(y^{\prime}(t) = 2y\) with the initial condition \(y(0) = 2\). The time step given is \(\Delta t = 0.5\).
02

Identify the function f(t, y)

The function to be used in Euler's method formula is found on the right side of the differential equation. In this case, it's \(f(t, y) = 2y\).
03

Calculate the first approximation \(u_1\)

Using Euler's method formula, let's calculate the first approximation \(u_1\). Here, \(t_0 = 0\), \(u_0 = y(0) = 2\), \(\Delta t = 0.5\). The formula is as follows: $$u_{n+1} = u_n + \Delta t \cdot f(t_n, u_n)$$ Replace the variables for the first approximation: $$u_1 = u_0 + \Delta t \cdot f(t_0, u_0) = 2 + 0.5 \cdot 2 \cdot 2$$ Compute the result: $$u_1 = 2 + 2 = 4$$
04

Calculate the second approximation \(u_2\)

Now, let's calculate the second approximation \(u_2\). Here, \(t_1 = 0.5, u_1 = 4, \Delta t= 0.5\). Use the same formula from the previous step: $$u_{n+1} = u_n + \Delta t \cdot f(t_n, u_n)$$ Replace the variables for the second approximation: $$u_2 = u_1 + \Delta t \cdot f(t_1, u_1) = 4 + 0.5 \cdot 2 \cdot 4$$ Compute the result: $$u_2 = 4 + 4 = 8$$
05

Conclusion

The first two approximations using Euler's method for the initial value problem \(y^{\prime}(t)=2y, y(0)=2 ; \Delta t=0.5\) are: $$u_1 = 4$$ $$u_2 = 8$$

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Key Concepts

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

Initial Value Problem
In mathematics, an initial value problem lays the foundation for needing solutions to differential equations. Essentially, it involves determining a function from its derivative (or rate of change) and an initial condition.
This initial condition typically specifies the value of the function at a given point, often called the starting point. For example, in this exercise, the initial condition is given as \( y(0) = 2 \).
This means that at time \( t = 0 \), the value of the function \( y \) is 2. Solving an initial value problem requires not just finding a family of solutions to the differential equation, but also picking out the one solution that fits the given initial condition.
  • Initial Value: Specifies where the function starts, such as \( y(0) = 2 \).
  • Objective: Find the specific function solution that satisfies both the differential equation and initial condition.
Differential Equations
Differential equations play a crucial role in modeling situations where quantities change over time. They involve equations that relate a function to its derivatives, expressing how the function changes. In this particular problem, the differential equation is \( y^{\prime}(t) = 2y \).
This equation suggests a relationship between the rate of change of \( y \), denoted as \( y^{\prime}(t) \), and the current value of \( y \) itself. Such equations are common in real-world scenarios, ranging from population growth to the decay of radioactive substances.
  • Significance: Illustrates how a function evolves over time.
  • Solution: Finding a function or set of functions that satisfy the equation throughout its domain.
The challenge with differentials is that while some have analytic solutions, others require numerical techniques like Euler's method.
Numerical Approximation
Numerical approximation, like Euler's Method, is vital when solving differential equations that don't have straightforward solutions. Euler鈥檚 Method allows us to calculate function values at discrete steps by finding slopes at points and moving a small step (\( \Delta t \)) along the slope.
In this example, we used a step size of \( \Delta t = 0.5 \) to approximate values \( u_1 \) and \( u_2 \). The method doesn't provide perfect solutions but offers a close enough approximation to understand how the function behaves over an interval.
  • Step Size \( \Delta t \): Determines precision and is the interval between successive points.
  • Process: Estimate the next point using current estimate and slope.
Euler's method is straightforward but might require smaller step sizes for better accuracy, yet larger sizes are computationally cheaper.

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Most popular questions from this chapter

Suppose the solution of the initial value problem \(y^{\prime}(t)=f(t, y), y(a)=A\) is to be approximated on the interval \([a, b]\). a. If \(N+1\) grid points are used (including the endpoints), what is the time step \(\Delta t ?\) b. Write the first step of Euler's method to compute \(u_{1}\). c. Write the general step of Euler's method that applies, for \(k=0,1, \ldots, N-1\).

A physiological model A common assumption in modeling drug assimilation is that the blood volume in a person is a single compartment that behaves like a stirred tank. Suppose that the blood volume is a four-liter tank that initially has a zero concentration of a particular drug. At time \(t=0,\) an intravenous line is inserted into a vein (into the tank) that carries a drug solution with a concentration of \(500 \mathrm{mg} / \mathrm{L} .\) The inflow rate is \(0.06 \mathrm{L} / \mathrm{min}\). Assume that the drug is quickly mixed thoroughly in the blood and that the volume of blood remains constant. a. Write an initial value problem that models the mass of the drug in the blood, for \(t \geq 0\) b. Solve the initial value problem and graph both the mass of the drug and the concentration of the drug. c. What is the steady-state mass of the drug in the blood? d. After how many minutes does the drug mass reach \(90 \%\) of its steady-state level?

U.S. population projections According to the U.S. Census Bureau, the nation's population (to the nearest million) was 281 million in 2000 and 310 million in \(2010 .\) The Bureau also projects a 2050 population of 439 million. To construct a logistic model, both the growth rate and the carrying capacity must be estimated. There are several ways to estimate these parameters. Here is one approach: a. Assume that \(t=0\) corresponds to 2000 and that the population growth is exponential for the first ten years; that is, between 2000 and \(2010,\) the population is given by \(P(t)=P(0) e^{n}\) Estimate the growth rate \(r\) using this assumption. b. Write the solution of the logistic equation with the value of \(r\) found in part (a). Use the projected value \(P(50)=439 \mathrm{mil}\) lion to find a value of the carrying capacity \(K\) c. According to the logistic model determined in parts (a) and (b), when will the U.S. population reach \(95 \%\) of its carrying capacity? d. Estimations of this kind must be made and interpreted carefully. Suppose the projected population for 2050 is 450 million rather than 439 million. What is the value of the carrying capacity in this case? e. Repeat part (d) assuming the projected population for 2050 is 430 million rather than 439 million. What is the value of the carrying capacity in this case? f. Comment on the sensitivity of the carrying capacity to the 40-year population projection.

The equation \(y^{\prime}(t)+a y=b y^{p},\) where \(a, b,\) and \(p\) are real numbers, is called a Bernoulli equation. Unless \(p=1,\) the equation is nonlinear and would appear to be difficult to solve-except for a small miracle. By making the change of variables \(v(t)=(y(t))^{1-p},\) the equation can be made linear. Carry out the following steps. a. Letting \(v=y^{1-p},\) show that \(y^{\prime}(t)=\frac{y(t)^{p}}{1-p} v^{\prime}(t)\). b. Substitute this expression for \(y^{\prime}(t)\) into the differential equation and simplify to obtain the new (linear) equation \(v^{\prime}(t)+a(1-p) v=b(1-p),\) which can be solved using the methods of this section. The solution \(y\) of the original equation can then be found from \(v\).

Consider the following pairs of differential equations that model a predator- prey system with populations \(x\) and \(y .\) In each case, carry out the following steps. a. Identify which equation corresponds to the predator and which corresponds to the prey. b. Find the lines along which \(x^{\prime}(t)=0 .\) Find the lines along which \(y^{\prime}(t)=0\) c. Find the equilibrium points for the system. d. Identify the four regions in the first quadrant of the xy-plane in which \(x^{\prime}\) and \(y^{\prime}\) are positive or negative. e. Sketch a representative solution curve in the xy-plane and indicate the direction in which the solution evolves. $$x^{\prime}(t)=-3 x+6 x y, y^{\prime}(t)=y-4 x y$$
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