Question

a first course in differential equations with modeling applications, 11th edition
dennis g. zill

Explanation:

🆕 New Concept Discovered: First-Order Linear Differential Equations
Solving equations of the form \( y' + P(x)y = f(x) \) using an integrating factor.

Step 1: Identify the standard form

A first-order linear differential equation has the standard form:

$$ \frac{dy}{dx} + P(x)y = f(x) $$

To solve any equation of this type, we must first divide through by any coefficient in front of \( \frac{dy}{dx} \) so that its coefficient is exactly \( 1 \). This allows us to correctly identify \( P(x) \), which is the key to finding our integrating factor.

Step 2: Find the integrating factor

Once we have identified \( P(x) \), we calculate the integrating factor, denoted by \( I(x) \) or \( \mu(x) \), using the formula:

$$ I(x) = e^{\int P(x) \, dx} $$

Multiplying the entire differential equation by this integrating factor collapses the left-hand side into a single derivative using the product rule in reverse:

$$ \frac{d}{dx} [I(x)y] = I(x)f(x) $$

Step 3: Integrate both sides

Finally, we integrate both sides with respect to \( x \):

$$ I(x)y = \int I(x)f(x) \, dx + C $$

We then solve for \( y \) by dividing both sides by \( I(x) \) to obtain the general solution.

Answer:

To solve a first-order linear differential equation of the form \( a_1(x) \frac{dy}{dx} + a_0(x)y = g(x) \):

1. Write in standard form: Divide by \( a_1(x) \) to get \( \frac{dy}{dx} + P(x)y = f(x) \).
2. Calculate the integrating factor: \( I(x) = e^{\int P(x) \, dx} \).
3. Multiply and integrate: Rewrite the equation as \( \frac{d}{dx}[I(x)y] = I(x)f(x) \) and integrate both sides to solve for \( y \).