Use the method of reduction of order to find a second independent solution of the given differential equation. t2y'' + 3ty' + y = 0, t > 0; y1(t) = t−1

I assume you mean , and not , since this doesn't satisfy the ODE.

Assume a second solution of the form , where is a function of . Then

Substituting into the ODE gives

where we omit the second term because it's already accounted for by .

5

step-by-step explanation:

x =±sqrt(1/3)

step-by-step explanation:

x+4/6x=1/x

multiply each side by 6x to get rid of the fractions

6x*(x+4/6x)=1/x*6x

distribute

6x^2 +4 = 6

subtract 4 from each side

6x^2 +4-4 = 6-4

6x^2 =2

divide by 6

6x^2/6 = 2/6

x^2 = 1/3

take the square root of each side

sqrt(x^2) = ±sqrt(1/3)

x =±sqrt(1/3)