Problem #8: Two linearly independent solutions of
xy" - (1+x)y' + y = 0
are y?(x) = 1 + x and y?(x) = e?.
(a) Find the Wronskian W(y?, y?) of y? and y?.
(b) Using the method of variation of parameters, we want to find a particular solution
yp of the following ODE
xy" - (1+x)y' + y = x²e²?
in the form yp(x) = u?(x)y?(x) + u?(x)y?(x) for some functions u? and u?. The
differential equation for u?(x) can be written as u?' = ... Type the right-hand side
of this equation into the answer box below.
(c) Solve the equation from part (b) to find the function u?.