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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 9 - Differential EquationsProblema 9.1.6
Capitolo 9, Problema 9.1.6

Explain why the graph of the solution to the initial value problem y'(t) = t²/(1 - t), y(-1) = ln 2 cannot cross the line t = 1.

Guida verificata passo dopo passo
1
Identify the differential equation given: \(y'(t) = \frac{t^2}{1 - t}\).
Notice that the denominator \(1 - t\) becomes zero at \(t = 1\), which means the right-hand side of the differential equation is undefined at \(t = 1\).
Since the derivative \(y'(t)\) is not defined at \(t = 1\), the solution \(y(t)\) cannot be extended continuously through \(t = 1\); this creates a vertical asymptote or a discontinuity in the slope.
The initial condition \(y(-1) = \ln 2\) is given at \(t = -1\), which is less than 1, so the solution is defined on an interval containing \(-1\) but cannot cross the point where the derivative is undefined, \(t = 1\).
Therefore, the graph of the solution cannot cross the vertical line \(t = 1\) because the differential equation does not allow the solution to be differentiable or continuous at that point.

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Existence and Uniqueness Theorem

This theorem states that if the function defining the differential equation and its partial derivative with respect to y are continuous near the initial condition, then a unique solution exists locally. It ensures the solution behaves predictably and helps identify points where the solution may fail to extend.
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Percorso guidato
06:11
Fundamental Theorem of Calculus Part 1

Domain of the Differential Equation

The domain is the set of t-values where the function defining y' is defined and continuous. Since y'(t) = t²/(1 - t) has a denominator that becomes zero at t = 1, the function is undefined there, creating a vertical asymptote or discontinuity that the solution cannot cross.
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Classifying Differential Equations

Behavior Near Singularities

When the differential equation has singularities (points where it is undefined), solutions often approach these points asymptotically but cannot cross them. At t = 1, the denominator zero causes the slope to become unbounded, preventing the solution curve from crossing this vertical line.
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Cases Where Limits Do Not Exist
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