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

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.

Answer 1

Answer:

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