The indicated function y1(x) is a solution of the given differential equation. use reduction of order or formula (5) in section 4.2, y2 = y1(x) e−∫p(x) dx y 2 1 (x) dx (5) as instructed, to find a second solution y2(x). y'' + 2y' + y = 0; y1 = xe−x

A differential equation shows the relationship between functions and their derivatives.

The equation of is:

The given parameters are:

The general equation is:

Compare the above equation to

Integrate:

We have:

The above equation becomes:

Substitute

Integrate

Evaluate the exponents

Cancel out common factors

Rewrite as:

Integrate

Set c to 0.

Hence, the equation of is:

Read more about differential equations at:

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First confirm that is a solution to the ODE,

We have

Substituting into the ODE gives

Suppose is another solution to this ODE. Then

and substituting these into the ODE yields

Let . Then the remaining ODE is linear in :

Multiply both sides by the integrating factor, , and condense the left hand side as a derivative of a product:

Integrate both sides with respect to and solve for :

Back-substitute and integrate both sides with respect to to solve for :

Back-substitute again to solve for :

already captures the solution , so the remaining one is

answer: the area of the parallelogram below is 42 square meters.

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