Use the method of reduction of order to find a second solution to t^2y' + 3ty' – 3y = 0, t> 0 given yı(t) = t y2(t) = preview give your answer in simplest form (ie no coefficients)

Let . Then

and substituting these into the ODE gives

Let , so that . Then the ODE is linear in , with

Multiply both sides by , so that the left side can be condensed as the derivative of a product:

Integrating both sides and solving for gives

Integrate again to solve for :

and finally, solve for by multiplying both sides by :

already accounts for the term in this solution, so the other independent solution is .

this is answer of 144 cubic meters