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 9.3.43a
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
Verified step by step guidance1
Rewrite the given differential equation \(e^{-\frac{y}{2}} y'(x) = 4x \sin(x^2) - x\) in the form \(\frac{dy}{dx} = e^{\frac{y}{2}} (4x \sin(x^2) - x)\) to isolate \(y'\) on one side.
Recognize that the equation is separable, so rearrange terms to separate variables: \(e^{-\frac{y}{2}} dy = (4x \sin(x^2) - x) dx\).
Integrate both sides: \(\int e^{-\frac{y}{2}} dy = \int (4x \sin(x^2) - x) dx\). For the left side, use substitution \(u = -\frac{y}{2}\); for the right side, split the integral into two parts and use substitution for \(\int 4x \sin(x^2) dx\).
After integrating, include the constant of integration \(C\) and write the implicit general solution relating \(y\) and \(x\).
Use the given initial conditions \(y(0) = 0\), \(y(0) = \ln(\frac{1}{4})\), and \(y(\sqrt{\frac{\pi}{2}}) = 0\) to solve for the constant \(C\) and verify the solution's consistency.
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7mKey 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 the product of a function of x and a function of y, allowing the variables to be separated on opposite sides of the equation. This enables integration with respect to each variable independently, facilitating the solution of the differential equation.
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Implicit Solutions
An implicit solution to a differential equation is a relation involving both x and y that defines y implicitly as a function of x. Unlike explicit solutions, implicit solutions may not isolate y on one side but still satisfy the differential equation and initial conditions.
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Initial Conditions and Particular Solutions
Initial conditions specify the value of the solution at a particular point, allowing determination of the constant of integration in the general solution. Applying these conditions yields a unique particular solution that fits the given problem context.
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Related Practice
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
Cooling time Suppose an object with an initial temperature of T₀ > 0 is put in surroundings with an ambient temperature of A, where A < T₀/2. Let t₁/₂ be the time required for the object to cool to T₀/2.
a. Show that t₁/₂ = −1/k ln((T₀ − 2A)/(2(T₀ − A))).
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
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.
Textbook Question
Stirred tank reaction A 100-L tank is filled with pure water when an inflow pipe is opened and a sugar solution with a concentration of 20 gm/L flows into the tank at a rate of 0.5 L/min. The solution is thoroughly mixed and flows out of the tank at a rate of 0.5 L/min.
c. At what time does the mass of sugar reach 95% of its steady-state level?
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