Domanda del libro di testo
Solving Initial Value Problems
Solve the initial value problems in Exercises 15–20.
t dy/dt + 2y = t³, t > 0, y(2) = 1
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Ch. 9 - First-Order Differential Equations
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 9, Problema 9.1.26
What integral equation is equivalent to the initial value problem y' = f(x), y(x₀) = y₀?
Guida verificata passo dopo passo1
Start with the given initial value problem (IVP): \(y' = f(x)\) with the initial condition \(y(x_0) = y_0\).
Recall that \(y' = \frac{dy}{dx}\) represents the derivative of \(y\) with respect to \(x\).
To find an equivalent integral equation, integrate both sides of the differential equation from \(x_0\) to \(x\): \(\int_{x_0}^{x} y'(t) \, dt = \int_{x_0}^{x} f(t) \, dt\).
By the Fundamental Theorem of Calculus, the left integral simplifies to \(y(x) - y(x_0)\), so we have \(y(x) - y_0 = \int_{x_0}^{x} f(t) \, dt\).
Rearranging, the integral equation equivalent to the IVP is \(y(x) = y_0 + \int_{x_0}^{x} f(t) \, dt\).
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Initial Value Problem (IVP)
An initial value problem specifies a differential equation along with a value of the unknown function at a particular point. For y' = f(x), y(x₀) = y₀, the solution must satisfy both the differential equation and the initial condition y(x₀) = y₀.
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Percorso guidato
05:03
Initial Value Problems
Fundamental Theorem of Calculus
This theorem connects differentiation and integration, stating that if a function is continuous, its integral can be used to recover the original function up to a constant. It allows rewriting the differential equation y' = f(x) as an integral equation involving the integral of f.
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06:11Fundamental Theorem of Calculus Part 1
Integral Equation Equivalent to IVP
The initial value problem y' = f(x), y(x₀) = y₀ can be expressed as y(x) = y₀ + ∫ₓ₀ˣ f(t) dt. This integral equation states that the solution y(x) equals the initial value plus the accumulated area under f from x₀ to x.
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08:02Parameterizing Equations
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