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 9.3.54a
[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.
Verified step by step guidance1
Rewrite the given differential equation \(y y'(t) = \frac{1}{2} e^{t} + t\) by recognizing that \(y'(t) = \frac{dy}{dt}\). This gives \(y \frac{dy}{dt} = \frac{1}{2} e^{t} + t\).
Separate variables by multiplying both sides by \(dt\) and dividing both sides by \(y\): \(y \, dy = \left( \frac{1}{2} e^{t} + t \right) dt\).
Integrate both sides: \(\int y \, dy = \int \left( \frac{1}{2} e^{t} + t \right) dt\). This will give you an implicit relation between \(y\) and \(t\).
Compute the integrals separately: \(\int y \, dy = \frac{y^{2}}{2} + C_1\) and \(\int \left( \frac{1}{2} e^{t} + t \right) dt = \frac{1}{2} e^{t} + \frac{t^{2}}{2} + C_2\). Combine constants into a single constant \(C\).
Solve for \(y\) explicitly by multiplying both sides by 2 and taking the square root: \(y = \pm \sqrt{e^{t} + t^{2} + C}\). Then express the general solution separately for \(y > 0\) and \(y < 0\) as \(y = +\sqrt{e^{t} + t^{2} + C}\) and \(y = -\sqrt{e^{t} + t^{2} + C}\) respectively.
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3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Separable Differential Equations
A separable differential equation can be written as a product of a function of y and a function of t, allowing variables to be separated on opposite sides of the equation. This enables integration with respect to each variable independently to find the general solution.
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Solving Separable Differential Equations
Integration of Both Sides
After separating variables, integrating both sides is essential to solve for y explicitly. This involves integrating functions of t on one side and functions of y on the other, often requiring techniques like substitution or recognizing standard integral forms.
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Handling Absolute Values and Sign Cases
When solving for y explicitly, the solution may involve absolute values due to integration of terms like 1/y. Distinguishing cases y > 0 and y < 0 ensures correct interpretation of the solution and avoids ambiguity in the sign of y.
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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
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
42–43. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.
a. Find the general solution of the equation.
e⁻ʸᐟ²y'(x) = 4x sin x² − x; y(0) = 0, y(0) = ln(1/4), y(√(π/2)) = 0
Textbook Question
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
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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.