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

y'(x) = −y + 2

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Rewrite the differential equation in the standard linear form: \(y'(x) + y = 2\).

Identify the integrating factor \(\mu(x)\), which is given by \(\mu(x) = e^{\int 1 \, dx} = e^{x}\).

Multiply both sides of the equation by the integrating factor to get: \(e^{x} y' + e^{x} y = 2 e^{x}\).

Recognize that the left side is the derivative of the product \(e^{x} y\), so write it as \(\frac{d}{dx} (e^{x} y) = 2 e^{x}\).

Integrate both sides with respect to \(x\): \(\int \frac{d}{dx} (e^{x} y) \, dx = \int 2 e^{x} \, dx\), then solve for \(y\) by dividing by \(e^{x}\) and adding the constant of integration.

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First-Order Linear Differential Equations

A first-order linear differential equation has the form y' + p(x)y = q(x). It involves the first derivative of the unknown function and can be solved using integrating factors or other standard methods. Recognizing this form is essential to apply the correct solution technique.

Integrating Factor Method

The integrating factor method involves multiplying the entire differential equation by a specially chosen function, usually e^(∫p(x)dx), to rewrite the left side as a derivative of a product. This simplifies solving the equation by allowing direct integration.

General Solution of Differential Equations

The general solution includes all possible solutions of a differential equation and typically contains an arbitrary constant. It combines the homogeneous solution (solving y' + p(x)y = 0) and a particular solution to the nonhomogeneous equation.

5–10. First-order linear equations Find the general solution of the following equations.y'(x) = −y + 2