The question is :
2x²y'' + 5xy' + y = x² - x;
y = c1x^(1/2) + c2x^(-1) + 1/15(x^2) - 1/6(x), (0,infinity)
The functions (x^-1/2) and (x^-1) satisfy the differential equation and are linearly independent since W(x^-1/2, x^-1)= ____?____ for 0
Answer:
The functions x^(-1/2) and x^(-1) are linearly independent since their wronskian is (-1/2)x^(-5/2) ≠ 0.
Step-by-step explanation:
Suppose the functions x^(-1/2) and x^(-1) satisfy the differential equation 2x²y'' + 5xy' + y = x² - x;
and are linearly independent, then their wronskian is not zero.
Wronskian of y1 and y2 is given as
W(y1, y2) = y1y2' - y1'y2
Let y1 = x^(-1/2)
y1' = (-1/2)x^(-3/2)
Let y2 = x^(-1)
y2' = -x^(-2)
W(y1, y2) =
x^(-1/2)(-x^(-2)) - (-1/2)x^(-3/2)x^(-1)
= -x^(-5/2) + (1/2)(x^(-5/2)
= (-1/2)x^(-5/2)
So, W(y1, y2) = (-1/2)x^(-5/2) ≠ 0
Which means the functions are linearly independent.
