Use the integration to find the general solution of the differential equation

Question

Use the integration to find the general solution of the differential equation  ​

Quizplus · Accepted Answer

The differential equation can be written as:

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

Using the integrating factor method, we first find the integrating factor:

$$I = e^{\int 1 dx} = e^x$$

Multiplying both sides of the differential equation by the integrating factor, we get:

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

Recognizing the left-hand side as the product rule of $(e^x y)'$, we can integrate both sides and solve for $y$:

$$\int (e^x y)' dx = \int xe^x dx$$

$$e^xy = xe^x - e^x + C$$

$$y = x - 1 + Ce^{-x}$$

This is the general solution to the differential equation, and the correct answer is choice D.

Use the Integration to Find the General Solution of the Differential