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33–42. Solving initial value problems Solve the following initial value problems.
p'(x) = 2/(x² + x), p(1) = 0
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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 9.1.39
33–42. Solving initial value problems Solve the following initial value problems.
y''(t) = teᵗ, y(0) = 0, y'(0) = 1
Guida verificata passo dopo passo1
Identify the given differential equation and initial conditions: \(y''(t) = t e^{t}\), with \(y(0) = 0\) and \(y'(0) = 1\).
Integrate the second derivative \(y''(t)\) once with respect to \(t\) to find the first derivative \(y'(t)\). This means computing \(y'(t) = \int t e^{t} \, dt + C_1\), where \(C_1\) is a constant of integration.
To integrate \(\int t e^{t} \, dt\), use integration by parts. Let \(u = t\) and $dv = e^{t} dt$, then find \(du\) and \(v\), and apply the formula \(\int u \, dv = uv - \int v \, du\).
After finding \(y'(t)\), use the initial condition \(y'(0) = 1\) to solve for the constant \(C_1\).
Integrate \(y'(t)\) with respect to \(t\) to find \(y(t)\), adding another constant of integration \(C_2\). Then use the initial condition \(y(0) = 0\) to solve for \(C_2\).
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Second-Order Differential Equations
A second-order differential equation involves the second derivative of a function. Solving such equations requires finding a function y(t) whose second derivative matches the given expression, often involving integration and applying initial conditions.
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07:39
Classifying Differential Equations
Initial Value Problems (IVPs)
An initial value problem specifies the values of a function and its derivatives at a particular point. These conditions allow us to determine the unique solution to a differential equation by solving for constants after integration.
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Percorso guidato
05:03Initial Value Problems
Integration of Non-Homogeneous Terms
When the differential equation includes a non-homogeneous term like te^t, solving involves integrating this term twice. Techniques such as integration by parts are often used to handle products of functions like t and e^t.
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Percorso guidato
05:44Divergence Test (nth Term Test)
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The general solution of a first-order linear differential equation is y(t) = Ce⁻¹⁰ᵗ − 13. What solution satisfies the initial condition y(0) = 4?
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27–30. Newton’s Law of Cooling Solve the differential equation for Newton’s Law of Cooling to find the temperature function in the following cases. Then answer any additional questions.
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Domanda del libro di testo
9–14. Growth rate functions Make a sketch of the population function P (as a function of time) that results from the following growth rate functions. Assume the population at time t = 0 begins at some positive value.
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