Domanda del libro di testo
17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.
y'(t) = cos² y, y(1) = π/4
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Ch. 9 - Differential Equations
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 9, Problema 9.2.13
12–16. Sketching direction fields Use the window [-2, 2] x [-2, 2] to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field is not needed.
y'(t) = 4−y, y(0) = −1
Guida verificata passo dopo passo1
Identify the differential equation given: \(y'(t) = 4 - y\). This means the slope of the solution curve at any point \((t, y)\) depends on the value of \(y\) at that point.
Set up a grid of points in the window \([-2, 2] \times [-2, 2]\) for \(t\) and \(y\). At each point \((t, y)\), calculate the slope \(y' = 4 - y\) to determine the direction of the solution curve.
At each grid point, draw a small line segment with slope equal to \(4 - y\). This collection of line segments forms the direction field, visually representing the behavior of solutions to the differential equation.
Use the initial condition \(y(0) = -1\) to locate the starting point on the \(t\)-\(y\) plane at \((0, -1)\). From this point, sketch a curve that follows the direction field, moving in the direction indicated by the slope segments.
Observe the general behavior of the solution curve: since \(y' = 4 - y\), when \(y\) is less than 4, the slope is positive, and when \(y\) is greater than 4, the slope is negative. This suggests the solution will approach the horizontal line \(y = 4\) as \(t\) increases.
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Direction Fields
A direction field is a graphical representation of a first-order differential equation showing the slope of the solution curve at various points. Each small line segment indicates the slope y' at that point (t, y), helping visualize the behavior of solutions without solving the equation explicitly.
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05:45
Understanding Slope Fields
Initial Value Problems
An initial value problem specifies a differential equation along with a starting point, such as y(0) = -1. This condition determines a unique solution curve passing through the given point, allowing us to sketch or find the particular solution that fits the initial data.
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Percorso guidato
05:03Initial Value Problems
Solving First-Order Differential Equations
Understanding how to solve or analyze first-order differential equations like y' = 4 - y is essential. This often involves recognizing the equation type (e.g., separable or linear), which helps predict solution behavior and sketch solution curves consistent with the direction field.
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06:06Solving Separable Differential Equations
Pratica correlata
Domanda del libro di testo
33–38. {Use of Tech} Solutions in implicit form Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one function, be sure to indicate which function corresponds to the solution of the initial value problem.
yy'(x) = 2x/(2 + y)², y(1) = −1
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Domanda del libro di testo
What is a separable first-order differential equation?
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Domanda del libro di testo
21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.
y'(t) = 3 + e⁻²ᵗ
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Domanda del libro di testo
17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.
y'(t) = eᵗʸ, y(0) = 1
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Domanda del libro di testo
17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.
y(t) = sec² t/(2y), y(π/4) = 1
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