Verify that y1 = x and y2 = x ln x are solutions to x 2y ′′ − xy′ + y = 0. b) use the wronskian to show that y1 and y2 are linearly independent. c) find the particular solution to the differential equation with initial conditions y(1) = 7, y′ (1) = 2

a. Substitute the given solutions and their derivatives into the ODE.

Both solutions satisfy the ODE.

b. The Wronskian determinant is

so the solutions are indeed independent.

c. The ODE has general solution . Then with the given initial conditions, the constants satisfy

So the ODE has the particular solution,

answer: it is a.,b.,d.

step-by-step explanation: