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.

hello,

b) 62.47% of the students are between 14 and 18 years old.

step-by-step explanation:

mean = 15 and standard deviation = 2

student between 14 to 18

formula used z-score

for age 14 , z score

using z-table , p(z> -0.5)=0.3085

for age 18 , z score

using z-table , p(z< 1.5)=0.9332

p(-0.5< z< 1.5)=0.9332-0.3085 = 0.6247

thus, 62.47% of the students are between 14 and 18 years old.

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step-by-step explanation: