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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 9.3.18
Capitolo 9, Problema 9.3.18

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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First, rewrite the differential equation in the form \( \frac{dy}{dt} = e^{ty} \) to clearly identify the variables involved.
Check if the equation is separable by trying to express it as a product of a function of \( t \) and a function of \( y \), i.e., \( \frac{dy}{dt} = g(t)h(y) \). In this case, observe that \( e^{ty} \) cannot be separated into a product of a function of \( t \) alone and a function of \( y \) alone.
Since the equation is not separable, the standard method of separation of variables does not apply here. Consider alternative methods such as substitution or recognizing the equation type.
If you attempt substitution, for example, let \( u = ty \), then express \( y \) and \( y' \) in terms of \( u \) and \( t \) to transform the equation into a potentially separable or solvable form.
After substitution, solve the resulting differential equation for \( u(t) \), then back-substitute to find \( y(t) \). Finally, apply the initial condition \( y(0) = 1 \) to determine the constant of integration.

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Separable Differential Equations

A differential equation is separable if it can be written as a product of a function of t and a function of y, allowing the variables to be separated on opposite sides of the equation. This form enables integration with respect to each variable independently to find the solution.
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Solving Separable Differential Equations

Initial Value Problems (IVP)

An initial value problem specifies the value of the unknown function at a particular point, providing a unique solution to a differential equation. Solving an IVP involves finding the general solution and then applying the initial condition to determine the constant of integration.
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Initial Value Problems

Integration Techniques for Exponential Functions

Solving differential equations involving exponential functions often requires integrating expressions like e^(t*y). Recognizing when to use substitution or other integration methods is essential to handle these integrals and find explicit solutions.
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Integrals of General Exponential Functions
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

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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

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.

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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.

(t² + 1)³yy'(t) = t(y² + 4)

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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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Domanda del libro di testo

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

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