Question

Solve the following differential equation. As you know, indefinite integrals are used to solve these equations and have an undetermined constant. In this exercise use C=0. dy/dx+2y=x

Answer 1

To solve the differential equation dy/dx + 2y = x, we use the method of integrating factors, apply integration by parts, and simplify the outcome considering the specified initial condition of C=0, obtaining the solution y(x) = (x/2 - 1/4)e^{-2x}.

Step-by-step explanation:

The differential equation given is dy/dx + 2y = x. This is a first-order linear differential equation and can be solved using the method of integrating factors.

1. Multiply the entire equation by an integrating factor that is calculated as e∫2dx, which simplifies to e2x.
2. Rewrite the equation as d/dx (y * e2x) = x * e2x.
3. Integrate both sides with respect to x.
4. The integration on the left side gives us y * e2x, and on the right side we need to use integration by parts for x * e2x.
5. The final solution will have the format y(x) = (x/2 - 1/4)e-2x + C, but since it's given that C = 0, the constant term can be omitted.

Thus, the particular solution to the differential equation is y(x) = (x/2 - 1/4)e-2x.