Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 9 - Differential Equations

Problema 9.R.3

Capitolo 9, Problema 9.R.3

2–10. General solutions Use the method of your choice to find the general solution of the following differential equations.
y′(t) + 2y = 6

Guida verificata passo dopo passo

Identify the type of differential equation. The given equation \(y'(t) + 2y = 6\) is a first-order linear ordinary differential equation.

Rewrite the equation in the standard linear form: \(y' + P(t)y = Q(t)\), where \(P(t) = 2\) and \(Q(t) = 6\).

Find the integrating factor (IF) using the formula \(\mu(t) = e^{\int P(t)\,dt}\). Here, calculate \(\mu(t) = e^{\int 2\,dt} = e^{2t}\).

Multiply both sides of the differential equation by the integrating factor \(e^{2t}\) to get \(e^{2t}y' + 2e^{2t}y = 6e^{2t}\). Notice that the left side is the derivative of \(e^{2t}y\).

Integrate both sides with respect to \(t\): \(\int \frac{d}{dt}(e^{2t}y)\,dt = \int 6e^{2t}\,dt\). Then solve for \(y(t)\) by dividing by \(e^{2t}\) and include the constant of integration.

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First-Order Linear Differential Equations

These are differential equations of the form y' + p(t)y = q(t), where y' is the derivative of y with respect to t. They can be solved using integrating factors or other methods to find the general solution that satisfies the equation for all t.

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Classifying Differential Equations

Integrating Factor Method

This technique involves multiplying the entire differential equation by an integrating factor, usually e^(∫p(t)dt), to rewrite the left side as the derivative of a product. This simplifies solving the equation by allowing direct integration to find y(t).

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Euler's Method

General Solution of Differential Equations

The general solution includes all possible solutions of a differential equation and typically contains an arbitrary constant. It represents the family of functions that satisfy the equation, encompassing both particular and homogeneous solutions.

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Solutions to Basic Differential Equations

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A second-order equation Consider the equation

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y(t) = C₁ tᵖ¹ + C₂ tᵖ²,

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