Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The general solution of the differential equation y'(t) = 1 is y(t) = t
Ch. 9 - Differential Equations
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 9, Problem 9.2.33a
33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
a. Approximate the value of y(T) using Euler’s method with the given time step on the interval [0,T].
y′(t) = -2y, y(0) = 1; Δt = 0.2, T = 2; y(t) = e⁻²ᵗ
Verified step by step guidance1
Identify the differential equation and initial condition: \(y'(t) = -2y\), with \(y(0) = 1\).
Set the time step \(\Delta t = 0.2\) and the interval from \(t=0\) to \(T=2\). Calculate the number of steps as \(n = \frac{T}{\Delta t} = \frac{2}{0.2} = 10\).
Recall Euler's method formula: \(y_{k+1} = y_k + \Delta t \cdot f(t_k, y_k)\), where \(f(t, y) = y'(t) = -2y\).
Start with the initial value \(y_0 = 1\) at \(t_0 = 0\). For each step \(k\) from 0 to 9, compute \(y_{k+1}\) using the formula: \(y_{k+1} = y_k + 0.2 \times (-2 y_k) = y_k - 0.4 y_k = y_k (1 - 0.4)\).
Repeat the calculation iteratively for all steps until you reach \(t = 2\). The final value \(y_{10}\) will be the Euler approximation of \(y(2)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Euler's Method
Euler's method is a numerical technique to approximate solutions of first-order differential equations. It uses a stepwise approach, updating the solution by moving along the slope given by the differential equation at each step. This method is especially useful when an exact solution is difficult to find.
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07:33
Euler's Method
Initial Value Problems (IVP)
An initial value problem specifies the value of the solution at a starting point, allowing the differential equation to be solved uniquely. Here, y(0) = 1 sets the initial condition, which is essential for applying Euler's method to approximate y(t) over the interval.
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05:03Initial Value Problems
Step Size and Interval in Numerical Methods
The step size (Δt) determines the increments at which the solution is approximated, affecting accuracy and computational effort. The interval [0, T] defines the domain over which the solution is computed. Smaller step sizes generally yield more accurate results but require more calculations.
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08:09Euler's Method Example 1
Related Practice
Textbook Question
38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.
a. Find the equilibrium solutions.
y′(t) = y(y - 3)
Textbook Question
[Use of Tech] Analysis of a separable equation Consider the differential equation yy'(t) = ½eᵗ + t and carry out the following analysis.
a. Find the general solution of the equation and express it explicitly as a function of t in two cases: y > 0 and y < 0.
Textbook Question
Direction field analysis Consider the first-order initial value problem y'(t)=ay+b,y(0)=A for t≥0 where a, b, and A are real numbers.
a. Explain why y=−b/a is an equilibrium solution and corresponds to a horizontal line in the direction field.
Textbook Question
38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.
a. Find the equilibrium solutions.
y′(t) = y(2 - y)
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Textbook Question
52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.
where P(t) is the population, for t ≥ 0, and r > 0 and K > 0 are given constants.
a. Verify by substitution that the general solution of the equation is P(t) = K/(1 + Ce⁻ʳᵗ), where C is an arbitrary constant.