Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 9 - Differential Equations

Problema 9.3.33

Capitolo 9, Problema 9.3.33

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.
y'(t) = 2t²/(y² − 1), y(0) = 0

Guida verificata passo dopo passo

Start with the given differential equation: $y'(t) = \frac{2t^{2}}{y^{2} - 1}$ and the initial condition $y(0) = 0$.

Rewrite the differential equation in separable form by expressing $y'(t)$ as $\frac{dy}{dt}$ and then separating variables: multiply both sides by $(y^{2} - 1) dt$ to get $(y^{2} - 1) dy = 2t^{2} dt$.

Integrate both sides: compute $\int (y^{2} - 1) \, dy$ on the left and $\int 2t^{2} \, dt$ on the right. This will give you an implicit relationship between $y$ and $t$.

After integration, include the constant of integration $C$ and use the initial condition $y(0) = 0$ to solve for $C$.

Write the implicit solution with the constant $C$ included. This implicit equation represents the solution to the initial value problem. Use graphing software to plot this implicit solution and identify which branch corresponds to the initial condition.

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Implicit Solutions of Differential Equations

An implicit solution defines a relationship between variables without explicitly solving for the dependent variable. In differential equations, implicit solutions may involve both variables intertwined, requiring techniques like implicit differentiation to analyze or graph the solution.

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Verifying Solutions of Differential Equations

Initial Value Problems (IVPs)

An initial value problem specifies a differential equation along with a condition that the solution must satisfy at a particular point. This condition helps identify the unique solution curve among multiple possible solutions represented by the implicit form.

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Percorso guidato

05:03

Initial Value Problems

Use of Technology for Graphing Solutions

Graphing software aids in visualizing implicit solutions, especially when explicit forms are difficult to obtain. It helps distinguish different solution branches and verify which corresponds to the initial condition, enhancing understanding of the solution's behavior.

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Percorso guidato

06:15

Graphing The Derivative

Pratica correlata

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.

p'(x) = 16/x⁹ - 5 + 14x⁶

144

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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.

z(x) = (z² + 4)/(x² + 16), z(4) = 2

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Domanda del libro di testo

21–24. Logistic equations Consider the following logistic equations. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed direction field is not needed. Assume t ≥ 0 and tP ≥ 0.

P′(t) = 0.05P − 0.001P²; P(0) = 10, P(0) = 40, P(0) = 80

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Domanda del libro di testo

5–10. First-order linear equations Find the general solution of the following equations.

y'(x) = −y + 2

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Domanda del libro di testo

Case 2 of the general solution Solve the equation y′(t) = ky + b in the case that ky + b < 0 and verify that the general solution is y(t) = Ceᵏᵗ − b/k.

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Domanda del libro di testo

5–16. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable.

e⁴ᵗy'(t) = 5

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