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

Use the formula:

∫xe^ (2x) dx=e^ (2x) (x/2-1/4).

Hint: Recognize this as a first-order linear differential equation and follow the general method for solving these.

y=x/2-1/4

Step-by-step explanation:

From exercise we have

C=0.

dy/dx+2y=x

Use the formula:

∫xe^ (2x) dx=e^ (2x) (x/2-1/4).

We know that a linear differential equation is written in the standard form:

y' + a (x) y = f (x)

we get that: a (x) = 2 and f (x) = x.

We know that the integrating factor is defined by the formula:

u (x) = e^{∫ a (x) dx}

⇒ u (x) = e^{∫ 2 dx} = e^{2x}

The general solution of the differential equation is in the form:

y=/frac{ ∫ u (x) f (x) dx + C}{u (x) }

⇒ y=/frac{ ∫ e^{2x}· x dx + 0}{e^{2x}}

y=/frac{e^{2x} (x/2-1/4) }{e^{2x}

y=x/2-1/4