Consider the differential equation x^2 y''-xy'-3y=0. if y1=x3 is one solution use redution of order formula to find a second linearly independent solution

Suppose is another solution. Then

Substituting these derivatives into the ODE gives

Let , so that

Then the ODE becomes

and we can condense the left hand side as a derivative of a product,

Integrate both sides with respect to :

Solve for :

Solve for :

So another linearly independent solution is .

10000

step-by-step explanation: divide by 2,