Question

To find a second solution y2(x) for the differential equation xy" y' = 0, given that y1 = ln x, what is the second solution?

Answer 1

The second solution for the differential equation xy" - y' = 0, given y1 = ln x, is y2(x) = C * ln(x), where C is an arbitrary constant.

Step-by-step explanation:

To find the second solution, we can use the method of reduction of order. Let's assume the second solution is of the form y2(x) = v(x) * y1(x), where v(x) is an unknown function.

Now, let's substitute this into the given differential equation: xy" - y' = 0. We get xy1'' + xy1' * v' - y1' * v = 0.

Since y1 = ln(x), y1' = 1/x and y1'' = -1/x^2. Substituting these values into the equation, we have -x * v = 0. Solving this equation gives the second solution y2(x) = C * ln(x), where C is an arbitrary constant.