8) use reduction of order to solve. one solution of homogeneo x2y" +7xy' +5y =x 1 x> 0 y1 = x here y1 is a solution of the corresponding homogeneous.

I suspect there's a typo in the question, because is *not* a solution to the corresponding homogeneous equation. We have and , so the ODE reduces to

Let , then and , and substituting these into the (homogeneous) ODE gives

which then admits the characteristic solutions and .

Now to find a solution to the non-homogeneous ODE. We look for a solution of the form or .

It doesn't matter which one we start with, so let's use the first case. We get derivatives and . Substituting into the ODE yields

Substitute , so that and

which is linear in , and we can condense the left side as the derivative of a product after multiplying both sides by :

Integrate to solve for :

Then multiply both sides by to solve for :

so we found another fundamental solution that satisifes this ODE.

step-by-step explanation:

y = | -2(-2) |     →     y = | 4 |     →     y = 4       ⇒   (-2, 4)

y = | -2(-1) |     →     y = | 2 |     →       y =   2       ⇒   (-1, 2)    

y = | -2(0) |     →     y = | 0 |     →       y = 0         ⇒   (0, 0)

y = | -2(1) |   c →     y = | -2 |     →     y =   2         ⇒   (1, 2)    

y = | -2(2) |     →     y = | - 4 |     →     y =   4         ⇒   (2, 4)  

answer: what is the question you're trying to ask?

step-by-step explanation: