Question

First-Order Linear EquationsSolve the differential equations in Exercises 1–14.xdy/dx + y = e ͯ, x \u003e 0

Accepted Answer

Rewrite the given differential equation in the standard linear form. The original equation is $$x \frac{dy}{dx} + y = e^x$$, with $$x \u003e 0. $$Divide both sides by $$x to $$isolate $$\frac{dy}{dx}$$:

$$\frac{dy}{dx} + \frac{1}{x} y = \frac{e^x}{x}. $$Identify the integrating factor (IF) for the linear differential equation. The IF is given by

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

where P(x$$) = \frac{1}{x}$$. So, calculate

$$\mu(x) = $$e^{\int \frac{1}$${x} \, dx} = $$e^{\ln|x|} = x$ (since x$$ \u003e 0$$). Multiply the entire differential equation by the integrating factor x$:

x$$ \frac{dy}{dx} + y = e^x$$

Notice this is the original equation, confirming the integrating factor is correct. Recognize that the left side of the equation is the derivative of the product x y$:

$$\frac{d}{dx} (x y) = e^x$$. Integrate both sides with respect to x$:

$$\int \frac{d}{dx} (x y) \, dx = \int e^x \, dx$$,

which simplifies to

x y = e^x + C$,

where C$ is the constant of integration. Finally, solve for y$ by dividing both sides by x$.

First-Order Linear Equations
Solve the differential equations in Exercises 1–14.

xdy/dx + y = e ͯ, x > 0

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Key Concepts

First-Order Linear Differential Equations

Integrating Factor Method

Exponential Functions and Their Properties

First-Order Linear EquationsSolve the differential equations in Exercises 1–14.xdy/dx + y = e ͯ, x > 0