2 Homogeneous second order ODE's with arbitrary coefficient: (40%)
Find the complete solutions of the homogeneous equations in form of Euler-Cauchy Equation or reduction
of order:
(a) $x^2y'' + 4xy' + 2y = 0$; $x > 0$, $y(1) = 0$, $y'(1) = 1$ (10%)
(b) $x^2y'' + 2xy' + \frac{1}{4}y = 0$; $x > 0$, $y(1) = 1$, $y'(1) = \frac{3}{2}$ (10%)
(c) $x^2y'' - 3xy' + 5y = 0$; $x > 0$, $y(1) = 2$, $y(e^{\frac{\pi}{2}}) = e^{\frac{\pi}{2}}$ (10%)
(d) Find general solution for: $x(x-1)y'' - xy' + y = 0$; (hint: find a simple $y_1(x)$ by observation, then
try order reduction) (10%)
